# Meta-meshing and triangulating lattice structures at a large scale

Qiang Zou<EMAIL_ADDRESS>Yunzhu Gao  Guoyue Luo  Sifan Chen  State
Key Laboratory of CAD&CG, Zhejiang University, Hangzhou, 310027, China

###### Abstract

Lattice structures have been widely used in applications due to their superior
mechanical properties. To fabricate such structures, a geometric processing
step called triangulation is often employed to transform them into the STL
format before sending them to 3D printers. Because lattice structures tend to
have high geometric complexity, this step usually generates a large amount of
triangles, a memory and compute-intensive task. This problem manifests itself
clearly through large-scale lattice structures that have millions or billions
of struts. To address this problem, this paper proposes to transform a lattice
structure into an intermediate model called meta-mesh before undergoing real
triangulation. Compared to triangular meshes, meta-meshes are very lightweight
and much less compute-demanding. The meta-mesh can also work as a base mesh
reusable for conveniently and efficiently triangulating lattice structures
with arbitrary resolutions. A CPU+GPU asynchronous meta-meshing pipeline has
been developed to efficiently generate meta-meshes from lattice structures. It
shifts from the thread-centric GPU algorithm design paradigm commonly used in
CAD to the recent warp-centric design paradigm to achieve high performance.
This is achieved by a new data compression method, a GPU cache-aware data
structure, and a workload-balanced scheduling method that can significantly
reduce memory divergence and branch divergence. Experimenting with various
billion-scale lattice structures, the proposed method is seen to be two orders
of magnitude faster than previously achievable.

###### keywords:

CAD; Geometric modeling; Lattice structures; Meta-meshing; Triangulation; GPU
computing

††journal: CAD

##  1 Introduction

With the advances in additive manufacturing, lattice structures have seen
increasing applications in fields like automotive, shipbuilding, and aerospace
industries [1, 2]. The network of interconnected struts in lattice structures
offers huge potential for design optimization, e.g., lightweight, high
strength, multifunctional [3, 4, 1, 5]. One of the essential elements in the
design and manufacturing pipeline of lattice structures is triangulation,
which discretizes the interconnected struts into triangles. Boundary
representation, like these triangles, is important to the visualization,
analysis, optimization, and fabrication steps in the pipeline [6, 7, 8, 9, 10,
11, 12, 13, 14, 15].

Existing methods for lattice structure triangulation basically fall under two
different strategies. The first strategy directly discretizes the struts into
triangles [16, 17, 18, 19, 20, 21, 22]. In the second strategy, a lattice
structure is first translated to other representation schemes [23, 24, 2, 25],
and then the triangulation is done on the translated version through, for
example, the marching cube algorithm [26].

The above two strategies can handle small-scale lattice structures but become
insufficient when large-scale lattice structures with millions or billions of
struts are considered. This is because lattice structures tend to have high
geometric complexity, and a tremendous number of triangles need to be
calculated and stored. Based on a back-of-the-envelope calculation, it would
require $>$100TB of memory and take hours to weeks to instantiate a cube of
1m3 volume with interconnected .1mm micro trusses [27]. In addition, the
diverse requirements of different applications give rise to the need for
meshes with varying resolutions for the same lattice structure. This need is
often met by starting over the triangulation algorithms again and again, which
is unacceptable.

To address these limitations, this paper proposes a new representation scheme
called meta-mesh for lattice structures. A meta-mesh is a collection of
connected vertices, circular/elliptical arcs, and cylindrical/conical faces
that make up a lattice’s boundary (see Sec. 4.1). It can be deemed as
“minimal” with regard to the number of vertices, arcs, and faces required to
represent a lattice structure. More importantly, it can serve as a reusable
base mesh for conveniently and efficiently triangulating lattice structures at
arbitrary resolutions—only arc subdivision is needed, and there is no need for
topology manipulation during triangulation. It can also generate a clean
triangulation that avoids redundant triangles for a given error tolerance.
Compared to the final fine triangular mesh, the number of elements in a meta-
mesh can be significantly reduced. (Note that the meta-mesh is not a
completely new notion, it has been occasionally and informally used in
computer graphics [28], and this paper extends it to the field of lattice
structure modeling.)

This paper also presents a CPU+GPU asynchronous pipeline to efficiently
generate meta-meshes for billion-scale lattice structures. It only takes
around 15 minutes to meta-mesh and then triangulate a billion-scale lattice
structure (on Nvidia RTX 3090). Achieving such a high performance is by no
means trivial, and the major challenges involved are as follows.

CPU-GPU data transfer bottleneck. It is well known that data transfer is
expensive and a major bottleneck in GPU computing [29]. For billion-scale
lattice structures, this bottleneck is even more severe since their file size
could be hundreds of GB or even TB [27]. To mitigate this problem, this work
designs a compressed representation of lattice structures to reduce the data
to be transferred. Furthermore, an asynchronous meta-meshing pipeline is
proposed to overlap data transfer and geometric computation, thereby hiding
the data transfer latency.

GPU-Lattice structural mismatch. Lattice structures consist of highly
interconnected struts, and therefore coupled geometric computing. By contrast,
GPUs are designed to leverage parallelism, requiring computing tasks to be
decoupled and independently processed on GPU cores. This mismatch between the
data structure and the computing architecture presents challenges to parallel
lattice structure triangulation. To solve this issue, we cast the coupled
strut-strut intersections (which underlies triangulation and meta-meshing) as
decoupled plane-strut intersections, which not only enable parallel computing
but also are much easier to calculate.

High warp divergence. Almost all existing parallel computing algorithms in CAD
[30, 31, 32] follow a thread-centric paradigm, which structures geometric
computing around threads [33]. Quite often, this way of working causes
unbalanced workloads among threads, warp divergence, and memory divergence,
leading to low parallelism. By contrast, this work shifts from thread-centric
to the very recent warp-centric paradigm, which structures geometric computing
around warps [34]. We achieve warp-centric meta-meshing by a novel GPU cache-
aware data structure, a coalesced-access memory layout, and a workload-
balanced scheduling method, which can distribute thread workloads evenly
within the same warp, thereby achieving high parallelism.

The main contributions of this paper are summarized as follows:

  * •

A meta-mesh representation scheme of lattice structures that is lightweight
and reusable to attain multiresolution triangulation;

  * •

A warp-centric GPU meta-meshing algorithm that can handle billion-scale
lattice structures in minutes, with features of decoupled geometric
calculation and low warp divergence; and

  * •

A CPU+GPU asynchronous meta-meshing pipeline that can hide CPU/GPU data
transfer latency and attain high triangulation throughput.

The remainder of this paper is organized as follows. Sec. 2 reviews the
literature, and Sec. 3 gives a very brief introduction to GPU computing. Secs.
4 and 5 elaborate the meta-meshing method and the triangulation method,
respectively. Validation of the method using a series of examples and
comparisons can be found in Sec. 6, followed by conclusions in Sec. 7.

##  2 Related work

In this section, we briefly discuss literature related to lattice
triangulation (Sec. 2.1) and GPU parallel computation (Sec. 2.2). Lattice
triangulation methods can be categorized as direct and indirect (further
divided into approximation- and isosurface extraction-based). The GPU parallel
computation part summarizes relevant methods, including GPU memory access,
warp-centric programming, and GPU-CPU cooperation.

###  2.1 Lattice triangulation.

Direct triangulation. This approach directly represents lattice structures as
triangles. There are two kinds of methods in this category. The first involves
replicating entirely meshed cell geometries (e.g., see [16, 17, 18]). The
second involves assembling meshed nodes and struts of cells separately (e.g.,
see [19, 20, 21]), which helps avoid complex node-strut intersections. While
current CAD tools can handle direct triangulation for small-scale lattice
structures, processing billion-scale lattice structures using this approach
presents challenges due to the model data explosion problem.

Indirect triangulation through isosurface extraction. The implicit modeling of
lattice structures has been widely employed due to its compact
representational advantages. With it comes research studies on the
triangulation of implicit lattice structures. Several methods have been
developed for this purpose, including the marching cube method [26] and the
marching tetrahedra method [25]. However, it should be noted that the use of
these algorithms may result in low-quality triangulations (e.g., redundant
facets), giving rise to concerns regarding representational accuracy.

Indirect triangulation via approximation. A recently emerging lattice
structure triangulation strategy is to mesh an approximating model of the
lattice structure, instead of the original model. As involved in [23, 24], the
convex hull nodes are used to approximate the spherical nodes of lattice
structures, then a coarse mesh can be obtained from the approximated lattice
structures. This can simplify the triangulation problem but results in error-
uncontrollable and low-quality meshes.

In summary, direct triangulation methods suffer from the disadvantage of poor
performance, while indirect triangulation methods (either based on iso-surface
extraction or approximation) result in low-quality meshes. Additionally, the
results from all methods are one-time and not reusable. To address these
shortcomings, this paper introduces a meta-meshing approach (Sec. 4).

###  2.2 GPU parallel computation

GPU computing has been extensively used in 3D modeling, including mesh
simplification [35], point-set triangulation [36], surface approximation [37],
among others. Since the proposed method is highly related to GPU computing, we
provide a summary of relevant GPU computing techniques and their recent
advancements.

GPU memory access. Due to the high latency and low bandwidth of GPU global
memory, memory access can cause performance bottlenecks as computations are
stalled by memory transactions. Various techniques have been employed to
address these bottlenecks, which include coalesced access, dedicated memories,
and data compression. Coalesced access effectively mitigates memory
transactions by modifying data layouts [38] or reorganizing threads [39].
However, achieving coalescing is challenged when problems necessitate random
access. Dedicated memories, such as constant memory [40] and texture memory
[41], offer low access latency but are generally limited in capacity. Data
compression is widely utilized to reduce both storage and access demands [42],
but this approach introduces additional computational overhead.

Warp-centric parallel programming. “Warp-centric” denotes a programming
paradigm that structures code around warps as units of computation [43]. This
approach is commonly utilized to reduce synchronization overhead [44] and
minimize warp divergence [45]. Implementing warp-centric programming involves
reorganizing the code so that the work is assigned to warps instead of threads
[46]. However, it’s essential to note that the warp-centric concept is
inherently tied to the GPU hardware. Code optimized for a specific warp size
or architecture may lack portability across diverse GPU platforms.

GPU-CPU cooperation. Research on cooperative GPU-CPU computing has been
explored to improve overall application performance. The current focus remains
on data transfer and work distribution between the GPU and CPU [47].
Techniques for transferring encompass pipelining [48] and data compression
[42]. Simultaneously, it is paramount to determine the most suitable device
for each part of an application to run on [47]. For the compute-intensive
tasks of lattice triangulation, it’s advisable to employ CPU-GPU cooperation
to improve the performance.

In summary, the aforementioned techniques offer valuable inspiration to
address the proposed challenges. However, it is important to note that there
is presently no direct GPU algorithm accessible for lattice triangulation.
Subsequent sections of this paper will integrate these techniques to solve the
GPU meta-meshing problem.

##  3 GPU Preliminaries

GPUs are typically structured with several stream multiprocessors (SMs), and
each SM is structured with multiple cores, as shown in Fig. 1. SMs have own
dedicated local resources (e.g., L1 cache called shared memory, registers) and
contain several blocks where each of them is a group of threads. All SMs have
access to some public resources, such as global memory, and L2 cache
positioned between L1 cache and global memory. Additionally, the GPU
integrates constant memory and texture memory as read-only memory caches.
Between the global memory and caches, data is transferred in unit called
“cache line”, a quantity (typically 128 bytes) determined by GPU hardware.

Figure 1: An overview of the GPU architecture.

Practically, the GPU divides computational resources into warps in SMs, which
contain multiple consecutive threads, e.g., 32 threads. Threads within a warp
execute the same instruction simultaneously. Executing different instructions
in a warp may result in warp divergence [49]. Moreover, memory accesses are
also issued per warp. If memory transactions cannot be coalesced into a single
cache line, the GPU has to perform multiple memory transactions to fetch
relevant data from the global memory to the caches to support the calculations
of all the threads in a warp. During such memory transactions, irrelevant
threads are forced to wait, resulting in low efficiency. In the worst case, a
memory transaction can only support a single thread’s data requirement, and
all the other threads in the warp are waiting. This problem is referred to as
memory divergence [50].

To fully leverage the parallel capabilities of the GPU, an application should
have the following features:

  * •

Global memory access reduction. Accessing global memory typically requires
several hundred GPU cycles. Therefore, minimizing global memory access is
critical to alleviate memory bottlenecks within GPU processing.

  * •

Memory and warp divergence reduction. Both divergences would produce much idle
time in GPU computing, leading to under-full utilization of cores and
therefore decreasing GPU gains. For such problems, coalescing memory
transactions and balancing the workload of warps are important for GPU
performance.

##  4 Meta-meshing

###  4.1 Meta-mesh definition and method overview

A meta-mesh is the collection of vertices, circular/elliptical arcs, and
trimmed cylindrical/conical surfaces, which make up the boundary of lattice
structures (Fig. 2a). Each strut intersects with several adjacent struts,
resulting in trimmed boundary surfaces. These intersection edges on the same
side of a strut, which are circular or elliptical arcs, are sequentially
interconnected, forming an arc loop as shown in Fig. 2b. The vertices are
located at the junctions of arcs. In such a case, lattice structures can be
represented with a minimal number of geometric entities like vertices, arcs,
and surfaces. With explicitly defined boundaries, the triangulation of lattice
structures at an arbitrary resolution can be efficiently obtained through arc
subdivision.

Figure 2: Illustrations of meta-meshes: (a) a single face; and (b) an arc
loop.

The notion of meta-mesh is simple by itself since it only decomposes a lattice
structure into bounded surfaces according to the natural strut-strut
intersections. However, meta-meshing a lattice structure is not trivial. This
is because each arc on a strut is derived from the strut-strut intersection
and needs to solve the overlap with other arcs to determine the connected
vertices. For a billion-scale lattice structure, it will take days to weeks to
generate the meta-mesh on an ordinary CPU, which is unacceptable. Accordingly,
utilizing GPU to process large-scale meta-mesh is preferred. As already noted
in the Introduction section, there are three major GPU meta-meshing challenges
involved in GPU meta-meshing, and new developments are needed. These
developments are summarized below:

  * •

High-throughput lattice structure data transfer. Two approaches are utilized
to improve the transfer speed. First, the lattice structures are compressed to
solve the CPU-GPU transfer bottleneck. Second, an asynchronous pipeline is
designed to hide the latency between GPU, CPU and disk. These two will be
detailed in Sec. 4.2.

  * •

Decoupled lattice structure computation. To overcome GPU-Lattice structural
mismatch, lattice structure decoupling is proposed to make GPU meta-meshing
available. By introducing auxiliary planes, interconnected struts are
decoupled into distinct elements so that meta-meshing can be parallelized on
the GPU. See Sec. 4.3.1 for more details.

  * •

Warp-centric lattice structure meta-meshing. To implement warp-centric meta-
meshing of lattice structures, a coalesced-access memory layout is designed to
optimize memory transactions. Then, the warp-centric task scheduling is used
to maximize GPU efficiency and performance gains. These related methods will
be illustrated in Secs. 4.3.2 and 4.3.3.

This work focuses on lattice structures with cylindrical and conical struts;
these struts also have tangential relationships with nodal spheres. Lattice
structures of this kind are among the most commonly used ones [22].

###  4.2 Data transfer of billion-scale lattice structures

Meta-meshing billion-scale lattice structures inevitably suffer from data
transfer bottlenecks. This leads to the GPU spending more than half of the
time waiting for data transfers rather than computing. Therefore, a compressed
lattice structure representation with controlled errors is proposed to
accelerate data transfers. Besides, a heterogeneous asynchronous pipeline is
designed to maximize bus bandwidth and throughput.

####  4.2.1 Compressive lattice structure representation

When two struts intersect at a node, the intersection curve is a truncated
elliptical arc as shown in Fig. 3, which can be expressed as:

$\begin{split}\begin{cases}x&=\bm{a}_{x}sin(t)+\bm{b}_{x}cos(t)+\bm{c}_{x}\\\
y&=\bm{a}_{y}sin(t)+\bm{b}_{y}cos(t)+\bm{c}_{y}\quad\forall
t\in[t_{1},t_{2}]\\\
z&=\bm{a}_{z}sin(t)+\bm{b}_{z}cos(t)+\bm{c}_{z}\end{cases}\end{split}$ (1)

where the vectors $\bm{a,b}$ are the ellipse’s semi-major axis and semi-minor
axis, respectively. $\bm{c}$ is the center of the ellipse. $t_{1}$ and $t_{2}$
are the parameter values of the arc’s two ends.

Figure 3: Illustration of an elliptical arc generated by intersecting two
struts.

To completely store this arc’s information, 11 parameters are needed (9 for
three vectors, 2 for the parameter domain). Assuming each parameter occupies a
4-byte floating-point number in GPU, the entire curve demands 44 bytes. For
billion-scale lattice structures, there are billions of intersection curves,
and the allocation for such large-scale geometric data in memory-limited GPU
is unfeasible.

To address this issue, a compressed storage scheme is proposed, which is
inspired by the Fixed-Point Representation method [51]. This compression
technique quantizes float-point values within a range onto a uniform integer
grid for efficient predictive compression. Based on this compression scheme,
the vectors $\bm{a,b}$ and $\bm{c}$ in Eq. (1) are first converted to
spherical coordinates, which is described by three parameters of radial
distance $r$, the polar angle $\theta$, and the azimuthal angle $\varphi$. The
two angle parameters inherently have fixed ranges ($[0,\pi]$ for polar angle,
$[0,2\pi]$ for azimuthal angle). For radial distance $r$, it can be easily
bounded by its minimum and maximum values (the length of the semi-major axis
lies in the interval $[R_{min},4R_{max}]$, the semi-minor axis in
$[0.1R_{min},R_{max}]$ and the center in $[0,R_{max}]$, where $R_{min}$ and
$R_{max}$ are minimum and maximum node radii, respectively)111It is possible
to have the lengths of the ellipse’s semi-major axes exceed the minimum and
maximum values used in this work. If such cases occur, we turn to the non-
compressed scheme (which of course would affect the overall computational
efficiency). Fortunately, such cases are rare, and they account for only a
small percentage out of all cases, less than 0.01% based on our statistics..
As a result, all these parameters can be coded into one or more integers.

Furthermore, some parameters do not need to be stored, such as the polar angle
$\theta_{b}$ and the azimuthal angle $\varphi_{b}$ of the semi-minor axis
$\bm{b}$. Since the semi-minor axis is perpendicular to both the strut’s
direction and the semi-major axis, $\theta_{b}$ and $\varphi_{b}$ can be
simply obtained by the cross-product. After removing redundant parameters, the
compression scheme is illustrated in Fig. 4b.

Figure 4: The storage of arcs in GPU global memory: (a) uncompressed storage;
and (b) compressed storage.

Error analysis and control. Since the proposed compression method is lossy,
the number of bits, used to code a floating-point number, needs to be
optimally determined to control compression error. For any point $\bm{p}$ on
the arc with parameter $t$, there is a corresponding point $\bm{p}^{\prime}$
after compression. The error $E$ between these two points can be measured by
the distance between $\bm{p}$ and $\bm{p}^{\prime}$:

$\begin{split}E=&||\bm{p}(t)-\bm{p}^{\prime}(t)||\\\
=&||(\bm{a}-\bm{a}^{\prime})sin(t)+(\bm{b}-\bm{b}^{\prime})cos(t)+(\bm{c}-\bm{c}^{\prime})||\end{split}$
(2)

where $\bm{a}$, $\bm{b}$ and $\bm{c}$ represent the elliptical parameters of
the uncompressed arc. Correspondingly, $\bm{a}^{\prime}$, $\bm{b}^{\prime}$
and $\bm{c}^{\prime}$ are the parameters after compression. To get the error
of $(\bm{a}-\bm{a}^{\prime})$, $(\bm{b}-\bm{b}^{\prime})$ and
$(\bm{c}-\bm{c}^{\prime})$, we estimate the error induced by compressing a
single vector as follows:

$\begin{split}&||\bm{a}-\bm{a}^{\prime}||\\\
=&\sqrt{(a_{x}-a_{x}^{\prime})^{2}+(a_{y}-a_{y}^{\prime})^{2}+(a_{z}-a_{z}^{\prime})^{2}}\\\
=&\sqrt{r^{2}+r^{\prime 2}-2rr^{\prime}(sin\theta
sin\theta^{\prime}cos(\varphi-\varphi^{\prime})+cos\theta
cos\theta^{\prime})}\\\
<&\sqrt{(r-r^{\prime})^{2}+2rr^{\prime}-2rr^{\prime}cos(\theta-\theta^{\prime})cos(\varphi-\varphi^{\prime})}\\\
\end{split}$ (3)

where $(a_{x},a_{y},a_{z})$ and
$(a_{x}^{\prime},a_{y}^{\prime},a_{z}^{\prime})$ are Euclidean coordinates of
$\bm{a}$ and $\bm{a}^{\prime}$. Then they are substituted by spherical
coordinates $(r,\theta,\varphi)$ and
$(r^{\prime},\theta^{\prime},\varphi^{\prime})$. Assuming $n$ bits for coding
radial distance $r$, $m$ bits for coding polar angle $\theta$, and $m+1$ bits
for the azimuthal angle $\varphi$. Because the semi-major axis’ length is
restricted to $[R_{min},4R_{max}]$, when encoding $r$ and $r^{\prime}$ using
$n$ bits, $(r-r^{\prime})$ has an error of $(4R_{max}-R_{min})/2^{n+1}$. For
$\theta\in[0,\pi]$ and $\varphi\in[0,2\pi]$, they are respectively encoded
using $m$ and $m+1$ bits. So $(\theta-\theta^{\prime})$ and
$(\varphi-\varphi^{\prime})$ both have the error of $\pi/2^{m+1}$. As a
result, the maximum value of $||\bm{a}-\bm{a}^{\prime}||$ is estimated as:

$\begin{split}||\bm{a}-\bm{a}^{\prime}||_{max}<R_{max}\sqrt{\cfrac{16}{2^{2n+2}}+\cfrac{32\pi^{2}}{2^{2m+2}}}\end{split}$
(4)

The error of $(\bm{b}-\bm{b}^{\prime})$ and $(\bm{c}-\bm{c}^{\prime})$ can be
estimated in the same way.

Using the distance between vectors before and after compression, the max error
of the elliptical arc can be estimated as:

$\begin{split}E\leq&||\bm{a}-\bm{a}^{\prime}||_{max}sin(t)+||\bm{b}-\bm{b}^{\prime}||_{max}cos(t)+||\bm{c}-\bm{c}^{\prime}||_{max}\\\
\leq&R_{Max}\left(\sqrt{\cfrac{17}{2^{2n+2}}+\cfrac{32\pi^{2}}{2^{2m+2}}}+\sqrt{\cfrac{1}{2^{2n+2}}+\cfrac{32\pi^{2}}{2^{2m+2}}}\right)\end{split}$
(5)

For triangulation applications, a max error less than 0.001$R_{max}$ is
sufficient according to our experience. By substituting 0.001$R_{max}$ to Eq.
(5), we can get 12 bits for the radial line $r$, 15 bits for the polar angle
$\theta$ and 16 bits for the azimuthal angle $\varphi$ to compress a vector.
For the other information like parameter range $t_{1}$ and $t_{2}$, they are
correspondingly compressed in 15 bits. After removing redundant parameters
like $\theta_{b}$ and $\varphi_{b}$, the storage of an individual arc can be
compressed from 352 bits to 128 bits.

####  4.2.2 Heterogeneous asynchronous pipeline

When the meta-mesh of a lattice structure is generated in the GPU, it is
transferred to the CPU for the subsequent triangulation task. There are good
reasons for using this CPU+GPU heterogeneous asynchronous pipeline. First, the
triangular mesh of lattice structures has a large amount of geometric data; if
using GPU for the triangulation, transferring so many triangles from GPU to
CPU cannot be done in a short time. Second, the use of meta-meshes can make
the triangulation simple and fast, thereby making it feasible to do
triangulation on the CPU. Therefore, using GPU only for meta-meshing and CPU
for triangulation improves the entire performance and avoids the bottleneck of
CPU-GPU data transfers.

To collaboratively utilize CPU, GPU, and IO devices, a heterogeneous
asynchronous pipeline is designed, which contains two levels of asynchrony.
The first asynchronously executes the meta-meshing of lattice structures in
GPU, the triangulation based on meta-mesh in CPU, and the final file writing.
With the batch-by-batch processing manner, when the CPU triangulates the
current batch, the GPU can meta-mesh the next batch while the I/O bus can
write the last batch to the disk, as shown in Fig. 5a. Another level of
asynchrony exists in data transfers between GPU and CPU. A larger batch is
further subdivided into small chunks suitable for GPU computation. While the
GPU is processing the current chunk, the next chunk is transferred to the GPU,
and the processing result of the last chunk is moved out GPU. This two-level
way of working can effectively hide GPU-CPU data transfer latency, as shown in
Fig. 5b.

Figure 5: The asynchronous pipeline: (a) GPU-CPU-Disk asynchronous execution;
and (b) GPU-CPU asynchronous transfers.

To effectively run the above pipeline, some parameters need to be set
properly. The batch needs to have a large size so that it can be written to
disk in blocks, but it is also limited by the I/O buffer size. To solve this
challenge, we provide the following equation to set a proper batch size:

$\begin{split}N_{batch}=\cfrac{S_{buffer}}{S_{triangle}*N_{strut}}\end{split}$
(6)

where $N_{batch}$ is the number of struts in a batch, $S_{buffer}$ is the
buffer size, $S_{triangle}$ is the triangle size, and $N_{strut}$ is the
number of triangles on a strut. Considering an example, an average of 20
triangles per strut ($N_{strut}=20$), 50 bytes per triangle
($S_{triangle}=50$), and a 4MB buffer size ($S_{buffer}=4,000,000$). Based on
Eq. (6), a batch can contain 4000 struts. To hide the latency of CPU-GPU data
transfer, we further divide each batch into 4 chunks, each of which contains
1000 struts. Like most GPU computation problems, there is no principled way to
determine the chunk size, and we set it to 4 through empirical studies. We
conducted experiments on a lattice structure dataset and examined different
chunk sizes (from 1 to 10), and eventually found that dividing each batch into
4 chunks gave the best performance.

###  4.3 Warp-centric GPU meta-meshing

To solve the GPU-Lattice structural mismatch challenge, a lattice structure
decoupling scheme is proposed, which converts interconnected lattice
structures into decoupled GPU-suitable parts. Then, coalesced memory layouts
and optimized warp-centric scheduling are presented to reduce memory and warp
divergence.

####  4.3.1 Decoupled lattice structure computation

The computation of meta-mesh involves determining the intersections between a
strut and its adjacent struts, which generate elliptical arcs. Since arcs can
overlap, their computation is interrelated, which is not in line with the
requirement of independent tasks for GPU threads. In this work, auxiliary
planes are introduced to decouple the computation on adjacent struts. By
replacing adjacent struts with planes, elliptical arcs can be computed in
parallel through strut-plane intersections.

Figure 6: The decoupled computing scheme for lattice structures: (a) strut-
plane intersection for ellipse computation; and (b) ellipse-plane intersection
to get arc loop.

Suppose a strut $S_{i}$ is defined by its two end vertices
$\bm{v}_{i}^{0},\bm{v}_{i}^{1}$ and the corresponding radii
$r_{i}^{0},r_{i}^{1}$. Further assume that this strut intersects with its
adjacent struts at $\bm{v}_{i}^{0}$. Following from Rossignac’s work [52], the
intersection curve of two struts is a planar conic section, and the plane
carrying the intersection curve can be easily calculated from the struts’
vertices and radii. In this work, we precompute such planes using the GPU and
the lattice’s topology graph (i.e., an adjacency list in the GPU). Let the
plane for two connecting struts $S_{i},S_{j}$ be denoted by $P_{i,j}$ (to be
called the auxiliary plane). We can then decouple the strut-strut intersection
problem between $S_{i},S_{j}$ into two simpler and independent strut-plane
problems, i.e., between $S_{i},P_{i,j}$ and between $S_{j},P_{i,j}$. Finding
the intersection (i.e., an ellipse) between a plane and a cylindrical/conical
strut is an elementary geometry problem, and the equations for calculating the
center $\bm{o}_{i,j}$, the semi-major axis $\bm{a}_{i,j}$, and the semi-minor
axis $\bm{b}_{i,j}$ are as follows (Chapter 1 of [53]):

$\begin{split}&\bm{o}_{i,j}=\cfrac{1}{2}(\bm{E}_{1}+\bm{E}_{2})\\\
&\bm{a}_{i,j}=\cfrac{1}{2}(\bm{E}_{1}-\bm{E}_{2})\\\
&\bm{b}_{i,j}=\lambda(\bm{a}_{i,j}\wedge\bm{n}_{i,j})\\\
with\quad&\bm{E}_{1}=\cfrac{\bm{n}_{i,j}\cdot(\bm{p}_{i,j}-\bm{F}_{1})}{\bm{n}_{i,j}\cdot
Rot(\bm{d_{i}^{\prime}},+\alpha)\bm{d}_{i}}\times
Rot(\bm{d_{i}^{\prime}},+\alpha)\bm{d}_{i}+\bm{F}_{1}\\\
&\bm{E}_{2}=\cfrac{\bm{n}_{i,j}\cdot(\bm{p}_{i,j}-\bm{F}_{2})}{\bm{n}_{i,j}\cdot
Rot(\bm{d_{i}^{\prime}},-\alpha)\bm{d}_{i}}\times
Rot(\bm{d_{i}^{\prime}},-\alpha)\bm{d}_{i}+\bm{F}_{2}\\\
&\bm{F}_{1}=\bm{v}_{i}^{0}+Rot(\bm{d_{i}^{\prime}},\alpha-\cfrac{\pi}{2})\bm{d}_{i}\times
r_{i}^{0}\\\
&\bm{F}_{2}=\bm{v}_{i}^{0}+Rot(\bm{d_{i}^{\prime}},\cfrac{\pi}{2}-\alpha)\bm{d}_{i}\times
r_{i}^{0}\\\
&\lambda=\sqrt{1-(\bm{a}_{i,j}\cdot\bm{d}_{i})^{2}/(cos\alpha\times\|\bm{a}_{i,j}\|)^{2}}\\\
\end{split}$ (7)

where operators “$\cdot$” and “$\wedge$” denote dot and cross product,
respectively. $\bm{n}_{i,j}$ and $\bm{c}_{i,j}$ are the orientation and
position of the plane $p_{i,j}$, respectively. $Rot(\bm{d^{\prime}},\alpha)$
is the matrix rotating a vector around axis $\bm{d}^{\prime}$ by angle
$\alpha$. $\alpha$ is strut’s conical angle given by
$arccos\left((r_{i}^{1}-r_{i}^{0})/||\bm{v}_{i}^{1}-\bm{v}_{i}^{0}||\right)$
(For cylindrical struts, $\alpha$ is 0). $\bm{d}_{i}$ is the normalized
direction vector from $\bm{v}_{i}^{1}$ to $\bm{v}_{i}^{0}$, and
$\bm{d_{i}^{\prime}}$ is the normalized version of
$\bm{d}_{i}\wedge\bm{n}_{i,j}$. By looping through intersected planes, all
ellipses can be obtained at the end section of the strut. This process can be
effectively parallelized on the GPU by assigning an ellipse into a thread (See
Fig. 6a).

After determining the intersected ellipses, there are overlaps among ellipses
on the strut. So one must compute the elliptical arc that is not overlapped by
other ellipses. That is to say, the connected vertices between ellipses need
to be calculated. To avoid time-consuming ellipse-ellipse intersections, an
ellipse is replaced by the auxiliary plane, and the connected vertices can be
obtained by the ellipse-plane intersection. Assuming ellipse $C_{i,j}$
overlaps with ellipse $C_{i,k}$, we can get the arc range $t$ on ellipse
$C_{i,j}$ with following equations:

$\begin{split}\begin{cases}\bm{v}=\bm{a}_{i,j}sin(t)+\bm{b}_{i,j}cos(t)+\bm{o}_{i,j}\\\
\bm{n}_{i,k}\cdot(\bm{v}-\bm{o}_{i,k})\leq 0\end{cases}\end{split}$ (8)
$\begin{split}cos^{-1}\cfrac{C}{\sqrt{A^{2}+B^{2}}}\leq
t&+tan^{-1}\cfrac{B}{A}\leq 2\pi-cos^{-1}\cfrac{C}{\sqrt{A^{2}+B^{2}}}\\\
where\quad&A=\bm{n}_{i,k}\cdot\bm{a}_{i,j}\\\
&B=\bm{n}_{i,k}\cdot\bm{b}_{i,j}\\\
&C=\bm{n}_{i,k}\cdot(\bm{o}_{i,k}-\bm{o}_{i,j})\\\ \end{split}$ (9)

where $\bm{v}$ is an arbitrary point on ellipse $C_{i,j}$, which does not
overlap with $C_{i,k}$. $\bm{n}_{i,k}$ is the normal of the plane where
$C_{i,k}$ lies on. With inter-thread communication (e.g., “shuffles”),
auxiliary planes can be broadcasted to other threads for the ellipse-plane
intersection, and the intersected vertices can be computed for each duet of
ellipses.

From Eq. (9), the elliptical arc can be obtained, which connected other arcs
at vertices $P_{1}$ and $P_{2}$, as shown in Fig. 6b. By merging these arcs
into an arc loop, the meta-mesh is acquired and then will be written to global
memory in a coalesced way, as will be explained in Sec. 4.3.2.

####  4.3.2 Coalesced-access memory layout

The meta-meshing results, a mass of arc loops, have to be temporarily stored
in global memory. Usually, the arcs within the same warp are positioned at
different addresses. Accessing these arcs leads to multiple memory
transactions per warp, which causes serious memory latency. To solve this
problem, a coalesced memory layout is proposed to coalesce the memory
transactions of warps for storing arcs.

The coalesced memory layout is handled by an index region and a data region.
The index region is a prefix-sum array, which records the starting address of
the meta-mesh for each strut. For example, the index region in Fig. 7 is [0,
5, 15, 32, 47…]. It means the first arc of strut 1 is at position 0 on the
data region and strut 2 is at position 5 and so on. By subtracting the index
from its next index, the number of arcs of a strut can be easily obtained.

The data region stores the meta-mesh consecutively, to make the warp access to
arcs can be as coalesced as possible. The two arc loops of each strut are
stored contiguously in memory. An arc loop consists of several arcs, which are
sequentially connected. The arc is the smallest unit in the meta-mesh and is
organized as Structure of Arrays (SoA) in the data region. In this
organization, the data region holds several one-dimensional arrays of the same
length, and then each arc is split into these arrays. To make memory access
coalescing in a cache line, the size of each arc element is defined by:

$\begin{split}S_{element}&=\cfrac{S_{cacheline}}{warpsize}\\\ \end{split}$
(10)

where $S_{element}$ is the size of each element in the SoA, and
$S_{cacheline}$ is the cache line size in the GPU. For example, given a cache
line size of 128 bytes and a warp size of 32, it is straightforward to
determine that each element is 4 bytes in size. Consequently, when threads in
a warp access a 4-byte arc element in the data region, it allows for the
coalescing of adjacent 32 elements into a single cache line.

In memory, the index region is small (a chunk of 1024 struts at most uses 8KB)
but is frequently accessed by the warp. Therefore, the prefix-sum array in the
index region can be saved in low-latency constant memory, which can reduce
memory latency in warps. Also, the prefix-sum array can be efficiently
obtained in GPU through parallel scanning algorithms [54].

Figure 7: The memory layout design for storing meta-meshes.

####  4.3.3 Optimized warp scheduling

Figure 8: Statistics of the number of arcs of the bone, gear and dragon
lattice structures from Fig. 11.

Lattice structures have complex connectivity, and the number of adjacent
struts that connect to any common strut is varied, leading to varied
quantities of strut-plane intersection for struts (or simply called
workloads). We collected struts from the bone, gear and dragon models in Fig.
11, and the statistics (Fig. 8) show that more than 70% of them are lower than
16 in the arc number. However, the warp size of a typical GPU is 32, using one
warp to process one strut would make nearly half of the threads in a warp
inactive, resulting in low parallelism.

To maximize warp utilization and gain massive parallelism, warp-centric
scheduling is utilized for the meta-meshing of the lattice structure. It
combines the meta-meshing tasks of multiple struts and assigns them to one
warp, making as many threads active as possible. For example, if two combined
struts both have workloads of 16, then a warp can be fully used to handle the
two struts. To achieve this objective, a sorting and scanning solution is
proposed. Sorting facilitates the positioning of combinable struts adjacent to
each other, and then scanning assigns struts that meet the combinable
requirement to warps. The specific sorting and scanning algorithms used are
Radix sorting and Blelloch scanning, which are both established and efficient
algorithms [49, 55].

There still exists some struts that have over 32 workloads, as shown in Fig.
8. Using a warp to process such struts demands a considerable amount of time.
This results in temporary underutilization of warps, where completed warps
must await uncompleted ones. For such a case, the scheduler employs multiple
warps to meta-mesh a single strut, and then synchronizes these warps through
shared memory. This scheduling approach enables efficient meta-meshing of
struts with more than 32 curves, thereby mitigating temporary warp
underutilization.

##  5 Triangulation

Based on the generated meta-mesh, triangulating lattice structures can be
simply done by subdividing the meta-mesh’s arcs. To make subdivided vertices
uniformly distributed on the arc and therefore obtain a high-quality
triangulation, the arc is projected to the circle edge of the strut’s end
section, and the subdivision process is implemented on the circle edge with
controllable chord error. Then these subdivided vertices are mapped to the
original arcs, as shown in Fig. 9. If we want to reuse the meta-mesh to
generate a triangulation at a different resolution, we only need to re-
subdivide the arcs according to the new chord error.

Figure 9: Illustration of subdividing arcs according to the chord error.

Assuming the chord error (measured in terms of the percentage of the radius)
is $CE$, and the parameter range of the arc is from $t_{1}$ to $t_{2}$, the
number of the subdivided vertices $N$ is given by:

$\begin{split}N=\left\lfloor\cfrac{t_{2}-t_{1}}{2cos^{-1}(1-CE)}\right\rfloor+1\\\
\end{split}$ (11)

where $\lfloor\rfloor$ is a floor function to get the integer part of the
value. The formula finds the minimum number of vertices, which satisfy the
specified chord error. In practical applications, multiple triangulation
results are required, like visualization. With uniform subdivision, we can
define the parameters $s_{i}$ on the original arcs by:

$\begin{split}s_{i}=\cfrac{t_{2}-t_{1}}{N}\times i+t_{1}\quad\quad
i=0,1,...,N\\\ \end{split}$ (12)

Using the above equation, all arcs on the two arc loops of a strut can be
subdivided. Then a strut can be easily triangulated by connecting these
vertices. Since the meta-mesh of each strut is independent, multiple CPU cores
can be leveraged to accelerate this process significantly. The time of the
triangulation overlaps with the meta-meshing in GPU and is fully hidden in the
pipeline, with no more extra idle time needed.

While triangulation is completed, there may still exist holes among adjacent
struts, as shown in Fig. 10a. The indication of holes lies in the special
circular arcs that are at the end of struts and have no intersection with
other struts, as shown in Figs. 2b and 3. Therefore, we can simply add a one-
bit flag to label such circular arcs, which enables automatic identification
of holes in the triangulation stage. With holes identified, we then take a
straightforward method to fill them: (1) computing a barycenter of the
vertices on the hole contour; (2) projecting the barycenter to its nodal
sphere; (3) connecting the contour vertices to the projected center. Assume a
hole contour is formed by vertices $\bm{v}_{0},\bm{v}_{1},...,\bm{v}_{M-1}$,
the barycenter $\bm{b}$ and the projected center $\bm{b}_{project}$ are given
by:

$\begin{split}\bm{b}&=\cfrac{\sum_{i=0}^{M-1}\bm{v}_{i}}{M}\\\
\bm{b}_{project}&=\cfrac{\bm{b}-\bm{o}}{||\bm{b}-\bm{o}||}\times r+\bm{o}\\\
\end{split}$ (13)

where $\bm{o}$ is the center of the nodal sphere with radius $r$. After
connecting vertices to the projected center, the hole is filled as shown in
Fig. 10b. It should be noted here that, this hole-filling method is chosen for
no particular reason but a feasible one. Our primary contribution lies in the
meta-meshing algorithm and pipeline presented in Sec. 4. If there are
alternative hole-filling methods, they can be readily integrated into the
pipeline without affecting the pipe’s effectiveness.

Figure 10: Hole filling: (a) extracting the hole contour; (b) filling the hole
by connecting the center and boundary vertices.

Based on the steps mentioned above, the triangulation process is implemented
using Algorithm 1. The algorithm accepts the meta-mesh and a list of chord
errors as the input, and outputs triangulation results with desired
resolutions. Starting from specified error $CE_{i}$, the first step subdivides
arcs into vertices (Line 5) and records vertices on the hole contour (Line 6).
Then, struts are triangulated by connecting subdivided vertices (Line 7).
Finally, it fills holes and writes the result triangles to the disk (Lines
9-10).

Algorithm 1 Triangulation

1:meta-mesh $MM$ and chord error list $CL=\\{CE_{1},CE_{2},\cdots,CE_{n}\\}$

2:lattice structure triangulation results

3:for each chord error $CE_{i}$ in $CL$ do

4: $T\leftarrow\emptyset$

5: $Holes\leftarrow\emptyset$

6: for each arcloop $AL$ in $MM$ do

7: $VL\leftarrow$ subdivide($AL,CE_{i}$)

8: $Holes\leftarrow Holes$ $\cup$ holesDetect($VL$)

9: $T\leftarrow T$ $\cup$ triangulate($VL$)

10: end for

11: $T\leftarrow T\cup$ fillHoles($Holes$)

12: write($T$) $\triangleright$ Output results to a file

13:end for

14:return

Figure 11: Models for case studies: (a) Venus with 0.5 billion struts; (b) Cow with 1 billion struts; (c) Stanford bunny with 1.4 billion struts; (d) Dragon with 1.9 billion struts; (e) Bone with 0.13 million struts; (f) Teddy bear with 1.5 million struts; (g) Kitten with 17 million struts; (h) Toy-connector with 3 million struts; (I) Gear with 17 million struts. Note that the rendered pictures here were based on simplified versions of billion-scale models for the sake of visual clarity; the actual billion-scale models are shown in the zoomed-in pictures. Table 1: Statistics of models generated by the meta-meshing method. Model |  Venus |  Cow |  Bunny |  Dragon |  Bone |  Teddy |  Kitten |  Connector |  Gear
---|---|---|---|---|---|---|---|---|---
Strut |  0.51B |  1.02B |  1.40B |  1.95B |  136K |  1.57M |  17.3M |  3.19M |  17.1M
Avg. strut len |  0.3mm |  0.05mm |  0.05mm |  0.05mm |  1.0mm |  1.0mm |  1.0mm |  0.1mm |  0.1mm
Radius |  0.05mm |  0.01mm |  0.01mm |  0.01mm |  0.2mm |  0.2mm |  0.2mm |  0.02mm |  0.02mm
Chord error |  5% |  3% |  5% |  5% |  2% |  2% |  2% |  6% |  6%
File size |  735G |  1,183G |  1,617G |  2,376G |  241M |  2.72G |  29.5G |  3.54G |  16.7G
Time |  617s |  995s |  1,091s |  1,868s |  0.118s |  1.20s |  15.7s |  1.93s |  9.36s
|  |  |  |  |  |  |  |  |

Figure 12: Comparisons of triangulation quality between MC, LSLT, and meta-meshing methods using the Bone model: (a) MC method; (b) LSLT method; (c) meta-meshing method. Table 2: Comparisons of MC, LSLT, and meta-meshing method. Model |  Strut |  Triangles (MC) |  Triangles (LSLT) |  Triangles (Meta-mesh) |  Time (MC) |  Time (LSLT) |  Time (Meta-mesh)
---|---|---|---|---|---|---|---
Bone |  136,391 |  262.4M |  8.37M |  4.78M |  11.2s |  19.9s |  0.118s
Teddy |  1,568,301 |  1,222M |  97.4M |  54.8M |  101.8s |  231s |  1.20s
Kitten |  17,331,861 |  Crashed |  1,092M |  595M |  N/A |  2,461s |  15.7s
|  |  |  |  |  |  |

##  6 Results and Discussion

To test the performance of the proposed method, a Nvidia RTX 3090 GPU paired
with an Intel Core i9-12900K CPU running at 5.20GHz is chosen. The GPU is
equipped with 82 streaming multiprocessors (SMs), 128 KB of L1 cache, 6 MB of
L2 cache, and a total of 24 GB of global memory. The method has been
implemented using C++ 17 and CUDA 12.2, operating within the Ubuntu 20.04
environment.

Based on this implementation, 9 case studies are to be presented to
demonstrate the effectiveness of the proposed method. Case studies 1-4 (Fig.
11a-d) considered four large lattice structures that have serval billion
struts. Case studies 5-7 (Fig. 11e-g) considered three small-scale lattice
structures, ranging from 0.1 to 10 million. They were used to carry out
comparisons with the state-of-the-art method [22]. The last two case studies
8-9 (Fig. 11h-i) used two real-world machine components. These two models were
used to verify whether the challenges were resolved. All these models were
generated by meshing boundary models using tetgen [56]. For billion-scale
models, which are too large to be tetrahedralized, we generated million-scale
models and then subdivided them to achieve the billion-scale counterparts.

###  6.1 Examples

Case study 1 (Fig. 11a) is a Venus model with bounding box of 160mm $\times$
80mm $\times$ 80mm. Case study 2 (Fig. 11b) considered a cow model with a
total of one billion struts. This model has a bounding box of 90mm $\times$
150mm $\times$ 150mm. Case study 3 (Fig. 11c) analyzed a bunny model with 1.4
billion struts. The size of the case is set to 120mm $\times$ 150mm $\times$
150mm. Case study 4 (Fig. 11d) is more complex, involving 2 billion struts and
a bounding box of 90mm $\times$ 140mm $\times$ 200mm. They are used to
evaluate the performance of billion-scale lattice structures. Moreover, these
four models have different strut radii and approximation errors.

Case study 5 (Fig. 11f) considered a bone model with 0.13 million struts. Case
study 6 (Fig. 11g) analyzed a teddy model, involving 1.5 million struts, and
case study 7 (Fig. 11h) is a kitten model characterized by 17.3 million
struts. They all have a strut radius of 0.2mm and chord error of 2%. The three
models are used for comparing with the marching cubes (MC) and LSLT method
[22]. We counted the triangulation runtime and the number of triangles
generated as summarized in Table 2. The triangulation quality of the three
methods has also been compared, as shown in Fig. 12. The proposed method is
seen to provide a significant improvement to the state of the art.

Case studies 8 (Fig. 11h) and 9 (Fig. 11i) considered two real-world machine
parts. Case study 8 is a toy connector with 3.2 million struts and a bounding
box of 16mm $\times$ 16mm $\times$ 32mm. Case study 9 analyzed a gear model,
involving 17.1 million struts, with a size of 80mm $\times$ 80mm $\times$
20mm. The radius (0.02mm) and chord error (6% of radius) are chosen for both
models. Both models can be downloaded from the GrabCAD library
(https://grabcad.com/library). To see whether the meta-meshing method resolves
data transfer and computational challenges mentioned in Sec. 1, we conducted
ablation experiments on case studies 8-9. The runtime of each pipeline stage
was collected to verify whether the transfer bottleneck had been overcome. For
GPU-Lattices structural mismatch, the traditional thread-centric method was
used for comparison with our meta-meshing method. The comparison results are
shown in Figs. 13 and 14.

In the above comparisons, the thread-centric meta-meshing scheme was
implemented by following the traditional thread-based algorithm design
paradigm. It takes a weighted graph representation of the lattice structure as
input and then allocates a thread for an arc calculation. During arc
calculation, every thread accesses the relevant strut parameters stored in the
global memory by itself, without considering other threads’ running status.
All scheduling and memory access optimization tasks are left to CUDA, if any.
The instructions used to calculate the specific intersection arc for two given
struts are the same as those used in our warp-centric scheme, so is the data
transfer between the CPU and the GPU.

Figure 13: Comparisons between the non-pipeline implementation and the
asynchronous pipeline with data compression. Figure 14: Comparisons between
thread-centric and warp-centric meta-meshing.

###  6.2 Discussions and limitations

Figure 15: Comparisons of computational efficiency between meta-meshing method
and LSLT method as the number of struts increases.

For all examples, the proposed method can efficiently and robustly generate
meta-meshes and triangular meshes. This method completes the triangulation of
the cow model with one billion struts within 17 minutes and exhibits excellent
scalability in different models. Even when processing the most complex dragon
model with 2 billion struts, triangulation can be completed within 30 minutes
(Table 1 and Fig. 15). By solving both data transfer and computational
bottlenecks on GPU, meta-meshing becomes even more efficient. Through a CPU-
GPU asynchronous pipeline and compression, the entire pipeline achieved a 2.5x
speedup (Fig. 13). Compared with the traditional thread-centric paradigm, the
GPU kernel execution in a warp-centric way is 35x faster than the thread-
centric way (Fig. 14).

In addition to high efficiency, the proposed method produces lightweight
triangular meshes. Compared with MC and LSLT, our method generates fewer
triangles under the same approximation error (Table 2). This is attributed to
the meta-mesh’s precise representation of struts. As can be seen from Fig. 12,
the MC method’s mesh contains many redundant triangle facets and is fairly
blocky. The LSLT method’s mesh does not align vertices between struts,
resulting in additional triangle faces. In the case of using a 2% chord error,
the number of triangles generated by the meta-meshing method is approximately
95% fewer than the MC method and about 50% fewer than the LSLT method. This
ratio is expected to decrease further if a larger chord error is used.

Some remarks should be noted here. For the LSLT method, it would be much
better to have a GPU version of it and then compare this version with the
proposed method. Unfortunately, there are two issues hindering the development
of the GPU version. First, we have done an extensive search on the internet
but found no publicly available codes for LSLT—the method’s authors have not
released their codes yet. Second, the original LSLT method was designated for
serial computing, and consequently converting its intrinsic serial logic into
a parallel logic is not straightforward; thus further development is required.
In these regards, a comparison with the GPU version of LSLT were not included
in the present work.

Despite the excellent performance of the meta-meshing method on large-scale
lattice structures, the entire pipeline often incurs startup time. For small-
scale data, using CPU processing alone proves more efficient and
straightforward. Fig. 15 illustrates the comparison between the meta-mesh and
LSLT methods regarding the processing time for lattice structures from hundred
to billion struts. It can be observed that the meta-meshing method requires a
startup time of 100ms-200ms in our hardware configuration. Therefore, in cases
where the number of struts is less than thousands, a CPU method may be easier
and more direct to triangulate the lattice structure. On the other hand, when
the number of struts reaches 1 million or even 1 billion, the meta-meshing
method has a much better performance than traditional CPU triangulation
methods.

##  7 Conclusion

This paper presents a new method called meta-meshing for solving the billion-
scale lattice structure triangulation problem. The main feature of this method
is its high efficiency, which can complete the triangulation of lattice
structures with a billion struts in minutes. This feature is essentially
achieved by using the meta-mesh representation, which is lightweight and can
conveniently triangulate lattice structures with arbitrary resolutions. By
resolving challenges in GPU-CPU data transfers, lattice-GPU mismatch, and warp
divergence, a high-performance CPU-GPU meta-meshing pipeline is developed. A
notable improvement over existing methods has been demonstrated in various
experiments.

It should be noted that the proposed method adopts a CPU+GPU collaborative
architecture. This architecture cooperatively utilizes GPU, CPU, and disk to
perform a task. However, the poor performance of a single device can impact
the efficiency of the entire pipeline. This is thus a serious limitation of
the proposed method. Moreover, the current meta-meshing method focuses solely
on the lattice structures with cylindrical and conical struts. There are still
other types of lattice structures that do not use cylinders and cones as
struts, such as Quador [52]. Extending the proposed method to handle more
complex lattice structures is an interesting improvement direction in the
future.

Another interesting improvement is to combine meta-meshing with the very
recent direct CAD modeling paradigm [57, 58, 59, 60]. As such, the user can
intuitively and freely edit a portion of a lattice structure, and then the
computer can promptly update its appearance, locally and adaptively. Without
meta-meshing, interaction-demanding applications such as direct modeling are
impossible.

## Acknowledgements

This work has been funded by NSF of China (No. 62102355), the “Pioneer” and
“Leading Goose” R&D Program of Zhejiang Province (No. 2024C01103), NSF of
Zhejiang Province (No. LQ22F020012), and the Fundamental Research Funds for
the Zhejiang Provincial Universities (No. 2023QZJH32).

## References

## References

  * [1] S. Liu, T. Liu, Q. Zou, W. Wang, E. L. Doubrovski, C. C. L. Wang, Memory-efficient modeling and slicing of large-scale adaptive lattice structures, Journal of Computing and Information Science in Engineering 21 (6) (2021) 061003.
  * [2] J. Ding, Q. Zou, S. Qu, P. Bartolo, X. Song, C. C. Wang, STL-free design and manufacturing paradigm for high-precision powder bed fusion, CIRP Annals 70 (1) (2021) 167–170.
  * [3] J. Liu, M. Xu, R. Zhang, X. Zhang, W. Xi, Progress of porous/lattice structures applied in thermal management technology of aerospace applications, Aerospace 9 (12) (2022) 827.
  * [4] K. J. Maloney, K. D. Fink, T. A. Schaedler, J. A. Kolodziejska, A. J. Jacobsen, C. S. Roper, Multifunctional heat exchangers derived from three-dimensional micro-lattice structures, International Journal of Heat and Mass Transfer 55 (9-10) (2012) 2486–2493.
  * [5] H. Yin, W. Zhang, L. Zhu, F. Meng, J. Liu, G. Wen, Review on lattice structures for energy absorption properties, Composite Structures 304 (1) (2023) 116397.
  * [6] Q. Zou, H.-Y. Feng, S. Gao, Variational direct modeling: A framework towards integration of parametric modeling and direct modeling in CAD, Computer-Aided Design 157 (2023) 103465.
  * [7] M. Li, C. Lin, W. Chen, Y. Liu, S. Gao, Q. Zou, Xvoxel-based parametric design optimization of feature models, Computer-Aided Design 160 (2023) 103528.
  * [8] X. Zou, S. B. Lo, R. Sevilla, O. Hassan, K. Morgan, The generation of 3D surface meshes for NURBS-enhanced FEM, Computer-Aided Design 168 (2024) 103653.
  * [9] Q. Zou, Length-optimal tool path planning for freeform surfaces with preferred feed directions based on poisson formulation, Computer-Aided Design 139 (2021) 103072.
  * [10] C. Su, X. Jiang, G. Huo, Q. Zou, Z. Zheng, H.-Y. Feng, Accurate model construction of deformed aero-engine blades for remanufacturing, The International Journal of Advanced Manufacturing Technology 106 (2020) 3239–3251.
  * [11] G. Luo, Q. Zou, A simple point-based iso-scallop tool path planning method for noisy point clouds with high robustness and controlled errors, Computer-Aided Design 163 (2023) 103560.
  * [12] Z. Wang, S. Liu, L. Liu, Q. Zou, X.-M. Fu, Computing smooth preferred feed direction fields with high material removal rates for efficient cnc tool paths, Computer-Aided Design 164 (2023) 103591.
  * [13] Q. Zou, J. Zhao, Iso-parametric tool-path planning for point clouds, Computer-Aided Design 45 (11) (2013) 1459–1468.
  * [14] Q. Zou, J. Zhang, B. Deng, J. Zhao, Iso-level tool path planning for free-form surfaces, Computer-Aided Design 53 (2014) 117–125.
  * [15] Q. Zou, Robust and efficient tool path generation for machining low-quality triangular mesh surfaces, International Journal of Production Research 59 (24) (2021) 7457–7467.
  * [16] M. Dumas, P. Terriault, V. Brailovski, Modelling and characterization of a porosity graded lattice structure for additively manufactured biomaterials, Materials & Design 121 (2017) 383–392.
  * [17] S. Vongbunyong, S. Kara, Rapid generation of uniform cellular structure by using prefabricated unit cells, International Journal of Computer Integrated Manufacturing 30 (8) (2017) 792–804.
  * [18] Y. Chen, 3D texture mapping for rapid manufacturing, Computer-Aided Design and Applications 4 (2007) 761–771.
  * [19] H. Wang, Y. Chen, D. W. Rosen, A hybrid geometric modeling method for large scale conformal cellular structures, in: Proceedings of the ASME International Design Engineering Techinical Conferences and Computers and Information in Engineering Conference, 2005, pp. 421–427.
  * [20] H. Wang, D. W. Rosen, Parametric modeling method for truss structures, in: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol. 36215, 2002, pp. 759–767.
  * [21] Y. Chen, A mesh-based geometric modeling method for general structures, in: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol. 42578, 2006, pp. 269–281.
  * [22] L. Chougrani, J.-P. Pernot, P. Veron, S. Abed, Lattice structure lightweight triangulation for additive manufacturing, Computer-Aided Design 90 (2017) 95–104.
  * [23] Y. Wu, A. Gupta, K. Kurzeja, J. Rossignac, CHoCC: Convex hull of cospherical circles and applications to lattices, Computer-Aided Design 129 (2020) 102903.
  * [24] C. S. Verma, B. Rankouhi, K. Suresh, A combinatorial approach for constructing lattice structures, Journal of Mechanical Design 142 (4) (2020) 041404.
  * [25] X. Ma, D. Z. Zhang, X. Yu, Z. Ren, S. Mao, X. Zheng, A novel design method for TPMS lattice structures with complex contour based on moving elements method, International Journal of Advanced manufacturing Technology 123 (1-2) (2022) 21–33.
  * [26] N. Stromberg, Optimal grading of TPMS-based lattice structures with transversely isotropic elastic bulk properties, Engineering Optimization 53 (11) (2021) 1871–1883.
  * [27] DARPA TRADES challenge problems, solidmodeling.org/trades-cp/, last accessed on 2024-1-25.
  * [28] P. Schmidt, J. Born, M. Campen, L. Kobbelt, Distortion-minimizing injective maps between surfaces, ACM Transactions on Graphics (TOG) 38 (6) (2019) 1–15.
  * [29] M. Zhu, Y. Zhuo, C. Wang, W. Chen, Y. Xie, Performance evaluation and optimization of HBM-enabled GPU for data-intensive applications, IEEE Transactions on Very Large Scale Integration (VLSI) Systems 26 (5) (2017) 831–840.
  * [30] A. Krishnamurthy, R. Khardekar, S. McMains, K. Haller, G. Elber, Performing efficient NURBS modeling operations on the GPU, in: Proceedings of the 2008 ACM Symposium on Solid and Physical Modeling, 2008, pp. 257–268.
  * [31] I. Hanniel, A. Krishnamurthy, S. McMains, Computing the Hausdorff distance between NURBS surfaces using numerical iteration on the GPU, Graphical Models 74 (2012) 255–264.
  * [32] W. Li, S. McMains, A sweep and translate algorithm for computing voxelized 3D Minkowski sums on the GPU, Computer-Aided Design 46 (2014) 90–100.
  * [33] Q. Xu, H. Jeon, K. Kim, W. W. Ro, M. Annavaram, Warped-slicer: Efficient intra-SM slicing through dynamic resource partitioning for GPU multiprogramming, International Symposium on Computer Architecture 44 (2016) 230–242.
  * [34] M. A. Awad, S. Ashkiani, R. Johnson, M. Farach-Colton, J. D. Owens, Engineering a high-performance GPU B-tree, in: Proceedings of the 24th Symposium on Principles and Practice of Parallel Programming, 2019, pp. 145–157.
  * [35] M. H. Mousa, M. K. Hussein, High-performance simplification of triangular surfaces using a GPU, Public Library of Science One 16 (8) (2021) e0255832.
  * [36] T.-T. Cao, A. Nanjappa, M. Gao, T.-S. Tan, A GPU accelerated algorithm for 3D Delaunay triangulation, in: Proceedings of the 18th Meeting of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games, 2014, pp. 47–54.
  * [37] M. H. Mousa, M. K. Hussein, Surface approximation using GPU-based localized fourier transform, Journal of King Saud University - Computer and Information Sciences 34 (4) (2022) 1431–1438.
  * [38] J. M. Cecilia, J. M. Garcia, A. Nisbet, M. Amos, M. Ujaldon, Enhancing data parallelism for ant colony optimization on GPUs, Journal of Parallel and Distributed Computing 73 (1) (2013) 42–51.
  * [39] N. Delbosc, J. L. Summers, A. I. Khan, N. Kapur, C. J. Noakes, Optimized implementation of the Lattice Boltzmann method on a graphics processing unit towards real-time fluid simulation, Computers & Mathematics with Applications 67 (2) (2014) 462–475.
  * [40] M. Fang, J. Fang, W. Zhang, H. Zhou, J. Liao, Y. Wang, Benchmarking the GPU memory at the warp level, Parallel Computing 71 (2018) 23–41.
  * [41] A. Monakov, A. Lokhmotov, A. Avetisyan, Automatically tuning sparse matrix-vector multiplication for GPU architectures, in: Proceedings of the 5th International Conference on High Performance Embedded Architectures and Compilers, 2010, pp. 111–125.
  * [42] V. Boyer, D. El Baz, M. Elkihel, Solving knapsack problems on GPU, Computers & Operations Resarch 39 (1) (2012) 42–47.
  * [43] P. Hijma, S. Heldens, A. Sclocco, B. Van Werkhoven, H. E. Bal, Optimization techniques for GPU programming, ACM Computing Surveys 55 (11) (2023) 1–81.
  * [44] A. Abdelfattah, A. Haidar, S. Tomov, J. Dongarra, Batched one-sided factorizations of tiny matrices using GPUs: challenges and countermeasures, Journal of Computational Science 26 (2018) 226–236.
  * [45] J. W. Bos, D. Stefan, Performance analysis of the SHA-3 candidates on exotic multi-core architectures, Cryptographic Hardware and Embedded Systems 6225 (2010) 279–293.
  * [46] N. Bell, M. Garland, Implementing sparse matrix-vector multiplication on throughput-oriented processors, in: Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis, 2009, pp. 1–11.
  * [47] K. Raju, N. N. Chiplunkar, A survey on techniques for cooperative CPU-GPU computing, Sustainable Computing-Informatics and Systems 19 (2018) 72–85.
  * [48] K. K. Yong, S. S. Othman Talib, Histogram optimization with CUDA, in: IEEE Industrial Electronics and Applications Conference (IEACon), 2016, pp. 312–318.
  * [49] CUDA C++ Programming Guide, https://docs.nvidia.com/cuda/cuda-c-programming-guide, last accessed on 2024-1-25.
  * [50] L. Wang, M. Jahre, A. Adileho, L. Eeckhout, MDM: The GPU memory divergence model, in: 53rd Annual IEEE/ACM International Symposium on Microarchitecture (MICRO), 2020, pp. 1009–1021.
  * [51] P. Lindstrom, M. Isenburg, Fast and efficient compression of floating-point data, IEEE Transactions on Visualization and Computer Graphics 12 (5) (2006) 1245–1250.
  * [52] A. Gupta, G. Allen, J. Rossignac, Quador: Quadric-of-revolution beams for lattices, Computer-Aided Design 102 (SI) (2018) 160–170.
  * [53] D. A. Brannan, M. F. Esplen, J. J. Gray, Geometry, Cambridge University Press, 2011.
  * [54] M. Harris, S. Sengupta, J. D. Owens, Parallel prefix sum (scan) with CUDA, GPU Gems 3 (39) (2007) 851–876.
  * [55] J. Han, B. Sharma, Learn CUDA Programming: A beginner’s guide to GPU programming and parallel computing with CUDA 10.x and C/C++, Packt Publishing, 2019.
  * [56] S. Hang, TetGen, a Delaunay-based quality tetrahedral mesh generator, ACM Transactions on Mathematical Software 41 (2) (2015) 1–36.
  * [57] Q. Zou, H.-Y. Feng, Push-pull direct modeling of solid CAD models, Advances in Engineering Software 127 (2019) 59–69.
  * [58] Q. Zou, H.-Y. Feng, A decision-support method for information inconsistency resolution in direct modeling of cad models, Advanced Engineering Informatics 44 (2020) 101087.
  * [59] Q. Zou, H.-Y. Feng, Variational b-rep model analysis for direct modeling using geometric perturbation, Journal of Computational Design and Engineering 6 (4) (2019) 606–616.
  * [60] Q. Zou, H.-Y. Feng, A robust direct modeling method for quadric b-rep models based on geometry–topology inconsistency tracking, Engineering with Computers 38 (4) (2022) 3815–3830.


# Revisiting the Black Brane Thermodynamic in Asymptotically Lifshitz
Spacetime in $\kappa$-Horndeski Gravity

Fabiano F. Santos1<EMAIL_ADDRESS>Eduardo Folco Capossoli2
<EMAIL_ADDRESS>Henrique Boschi-Filho1<EMAIL_ADDRESS>1Instituto de Física, Universidade Federal do Rio de Janeiro, 21.941-972, Rio
de Janeiro - RJ, Brazil
2Departamento de Física and Mestrado Profissional em Práticas de Educação
Básica (MPPEB), Colégio Pedro II, 20.921-903, Rio de Janeiro - RJ, Brazil

###### Abstract

In this article, we investigate the thermodynamics of an asymptotically
Lifshitz spacetime within $\kappa$-Horndeski gravity where the
$\kappa$-algebra will be implemented in the kinetic term of this theory. In
our investigation, we considered the black brane solutions in
(3+1)-dimensions, which is a solution of $\kappa$-Horndeski theory for any
value of critical exponent. In the canonical Ensemble, we explore the
evolution of thermodynamics quantities in function of critical exponents and
of the $\kappa$-algebra.

##  I Introduction

Since its discovery in 1974, Horndeski’s gravity has aroused great interest
ranging from applications in the field of cosmology to black holes
Heisenberg:2018vsk ; Horndeski:1974wa ; Bruneton:2012zk ; Brito:2018pwe .
However, a case of particular interest is the study of static black hole
solutions Cisterna:2014nua ; Cisterna:2016vdx  with asymptotically anti-de
Sitter behavior. This Anti-de Sitter (AdS) spacetime played an extremely
important role in the holographic duality known as AdS/CFT Gibbons:2011sg ;
Maldacena:1997re . In recent years, the gauge/gravity duality has offered
remarkable support for the dynamics of systems strongly coupled to condensed
matter systems Hartnoll:2009sz ; Sachdev:2011wg  in the study of the universal
limits of transport coefficients, including viscosity/entropy ratio
Kovtun:2003wp ; Kovtun:2004de , which is conjectured based on the "bottom up"
models. Some examples of violation of this limit were presented by
Feng:2015oea ; Brito:2019ose ; Sadeghi:2018vrf , however, in Feng:2015oea ;
Brito:2019ose ; Santos:2020egn  considerations of the AdS/CFT correspondence
were presented in the context of Horndeski’s gravity.

However, the recent discovery of event GW170817 of the merger of the two
neutron stars, has imposed strong restrictions on sectors of Horndeski’s
gravity Oikonomou:2020sij  one of these sectors is the Jhon sector of
Horndeski’s gravity, this sector derives from a special subclass known as
theories F4 Charmousis:2011bf ; Charmousis:2011ea . Horndeski’s gravity
Heisenberg:2018vsk ; Horndeski:1974wa ; Bruneton:2012zk ; Brito:2018pwe , is
characterized by a single theory of scalar tensors with second order field
equations and second order energy moment tensor Heisenberg:2018vsk ;
Horndeski:1974wa ; Cisterna:2014nua ; Bravo-Gaete:2013dca ; Anabalon:2013oea ;
Rinaldi:2012vy . The Lagrangians associated with this theory produce second
order equations of motion Anabalon:2013oea ; Deffayet:2011gz ; Rinaldi:2016oqp
; Cisterna:2017jmv . In addition, the theory includes four arbitrary functions
of the scalar field and its kinetic term Charmousis:2011bf ; Charmousis:2011ea
; Gleyzes:2013ooa ; Zumalacarregui:2013pma ; Starobinsky:2016kua . Within the
scope of the construction of black holes this theory has shown that it is
possible to work on the development of Hawking-Page phase transitions at a
critical temperature Anabalon:2013oea ; Rinaldi:2012vy ; Rostami:2019xrx .
However, in the context of high energy particle collisions, we have the
creation of tiny black holes for the primitive Universe, this can leave stable
remains instead of completely evaporating as a result of the Hawking radiation
Nakama:2018lwy . In addition, at scales comparable to quantum gravity near
TeV, the LHC (Large Hadron Collider) can produce a black hole every second.
Black holes are well understood objects, when their mass exceeds Planck’s
fundamental mass Mp$\sim$ TeV. In this case, if the mass of the black hole
approaches that of Planck (MBH$\approx$Mp), we have that black holes become
"sticky" Dimopoulos:2001hw  and its properties end up becoming complex in this
regime. This obstacle can be ignored by taking semi-classical arguments, that
is, when we are in the regime where MBH$>>$Mp.

Despite the wide variety of studies involving black holes Brito:2018pwe ;
Anabalon:2013oea ; Rinaldi:2012vy ; Rostami:2019xrx  Hawking provided that
black holes for an AdS geometry have a minimum temperature, which occurs when
its size is in the order of the characteristic radius of the AdS space
Hawking:1982dh . On the other hand, for larger black holes their deviation
from temperature measured at infinity to red is large. This fact showed that
these black holes have a specific positive heat and can be in stable
equilibrium with thermal radiation at a fixed temperature. In addition,
another implication is that the canonical Ensemble exists for the space
asymptotically AdS Hartnoll:2009sz ; Hawking:1982dh ; Ong:2015fha .

Recent investigations regarding black hole thermodynamics have been matched in
Anabalon:2013oea ; Baake:2020tgk  where studies involving a gravitational
soliton were considered as a background to investigate finite Euclidean
action. In addition, in Baake:2020tgk  thermodynamics for BTZ-like black holes
was discussed by both the Euclidean and Cardy methods. On the other hand, we
can explore the thermodynamics of black holes through a very elegant and
robust argument, in which we have that the horizons correspond to thermally
mixed states Hartnoll:2009sz ; Susskind:2005js . However, for a semi-classical
regime, the volume theory partition function (accompanied by asymptotic
boundary conditions of the AdS, must provide an equivalence with the large
field theory partition function) as an integral path over the metrics. As
addressed by Oikonomou:2020sij  one of the reasons that put many corrected
cosmological models to the test is the fact that the vast majority of them
produce massive gravitons primarily. One of these models that we have already
commented on is the model involving the Jhon sector term of Horndeski’s
gravity. In our analyzes, we will compute the thermodynamics quantities
through the canonical Ensemble. For this, we need to render the finite action.
A way to perform this is construct a boundary systems take into account a
Gibbons-Hawking surface term to the Horndeski gravity Santos:2021orr ;
Li:2018rgn .

In our studies, we will analyze the thermodynamics of a black brane in
asymptotically Lifshitz spacetime in $\kappa$-Horndeski Gravity. Thus, we
propose a new description of the Horndeski gravity deforming the kinetic term
of Feng:2015oea ; Brito:2019ose ; Santos:2020egn ; Charmousis:2011bf ;
Charmousis:2011ea ; Starobinsky:2016kua  inspire in the works of
Tsallis:1987eu ; daCosta:2020mbf ; Ernesto  specially in daCosta:2020mbf , we
introduced the $\kappa$-algebra in the term
$\nabla^{\mu}\phi\nabla^{\nu}\phi$. Besides, the kinetic terms associated to
the boundary counter-term will be deformed. The motivation to implement this
modification is for obtain a solution without constraints over the critical
exponent, but solutions with particular values of the critical exponent in
Horndeski gravity were present in Brito:2019ose ; Bravo-Gaete:2013dca . This
work is summarized as follows: In Sec.$\sim$II, we present
$\kappa$-Horndeski’s Theory. In Sec.$\sim$III, we address the issue of finding
the ansatz black brane in four-dimensions. In Sec.$\sim$IV, we explore the
canonical set and analyze thermodynamic quantities. Finally, in Sec.$\sim$V,
we present our conclusions.

##  II $\kappa$-Horndeski Theory

In this session, we will match the Jhon sector of Horndeski’s gravity
Charmousis:2011bf ; Charmousis:2011ea ; Starobinsky:2016kua . This model is
expressed by the following action

$\displaystyle I[g_{\mu\nu},\phi]=\int{d^{4}x\sqrt{-g}\mathcal{L}_{H}}+S_{m}.$
(1) $\displaystyle\mathcal{L}_{H}=k(R-2\Lambda)-\frac{1}{2}(\alpha
g_{\mu\nu}-\gamma
G_{\mu\nu})\nabla^{\mu}_{\kappa}\phi\nabla^{\nu}_{\kappa}\phi$ (2)

Here, we adopt $k=(16\pi G)^{-1}$ where
$\nabla^{\nu}_{\kappa}=\sqrt{\Theta_{\kappa}(r)}\nabla^{\nu}$ with
$\Theta_{\kappa}(r)=\kappa^{2}\Theta(r)$ is a dimensionless function,
$\kappa$-deformed derivative is associated to the Kappa statistics
daCosta:2020mbf , we explicit this function with more details in
Sec.$\sim$III. $S_{m}$ is used to describe a perfect fluid. In this action,
the field has a dimension of $(mass)^{2}$ and the parameters $\alpha$ and
$\gamma$ control the strength of the kinetic couplings, $\alpha$ is
dimensionless and $\gamma$ has a dimension of $(mass)^{-2}$. Another
characteristic of the term
$\nabla^{\mu}_{\kappa}\phi\nabla^{\nu}_{\kappa}\phi$ is that it is a second-
order covariant derived from the scalar field. Furthermore, action is
invariable under the displacement symmetry $\phi\to\phi+constant$ and under
the discrete transformation $\phi\to-\phi$. Thus, the Einstein-Horndeski field
equations can be written formally by varying the action (1) $\delta
S[g_{\mu\nu},\phi]$ providing

$G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{\Theta_{\kappa}(r)}{2k}T_{\mu\nu},$ (3)

where $T_{\mu\nu}=\alpha T^{(1)}_{\mu\nu}+\gamma T^{(2)}_{\mu\nu}$ the energy-
momentum tensors $T^{(1)}_{\mu\nu}$ and $T^{(2)}_{\mu\nu}$ take the following
form

$\displaystyle
T^{(1)}_{\mu\nu}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\nabla_{\lambda}\phi\nabla^{\lambda}\phi$
$\displaystyle T^{(2)}_{\mu\nu}=\frac{1}{2}\nabla_{\mu}\phi\nabla_{\nu}\phi
R-2\nabla_{\lambda}\phi\nabla_{(\mu}\phi
R^{\lambda}_{\nu)}-\nabla^{\lambda}\phi\nabla^{\rho}\phi
R_{\mu\lambda\nu\rho}$
$\displaystyle-(\nabla_{\mu}\nabla^{\lambda}\phi)(\nabla_{\nu}\nabla_{\lambda}\phi)+(\nabla_{\mu}\nabla_{\nu}\phi)\Box\phi+\frac{1}{2}G_{\mu\nu}(\nabla\phi)^{2}$
$\displaystyle-
g_{\mu\nu}\left[-\frac{1}{2}(\nabla^{\lambda}\nabla^{\rho}\phi)(\nabla_{\lambda}\nabla_{\rho}\phi)+\frac{1}{2}(\Box\phi)^{2}-(\nabla_{\lambda}\phi\nabla_{\rho}\phi)R^{\lambda\rho}\right].$

And the scalar field equation is given by

$\nabla_{\mu}[(\alpha g^{\mu\nu}-\gamma
G^{\mu\nu})\Theta_{\kappa}(r)\nabla_{\nu}\phi]=0.$ (4)

##  III Black brane solutions in asymptotically Lifshitz spacetime

In our case, for Einstein-Horndeski gravity, we consider the following Ansatz
for a four-dimensional black brane of the form Brito:2019ose ; Santos:2020egn
:

$ds^{2}=L^{2}\left(-r^{2z}f(r)dt^{2}+r^{2}(dx^{2}+dy^{2})+\frac{dr^{2}}{r^{2}f(r)}\right).$
(5)

The spherically symmetrical static configurations of certain Galileons have an
invariance of change, these were first proposed to admit a no-hair theorem
Brito:2019ose ; Bravo-Gaete:2013dca . In addition, the no-hair theorem for
Galileons requires that the square of the conserved current $J_{\mu}=(\alpha
g_{\mu\nu}-\gamma G_{\mu\nu})\nabla^{\nu}_{\kappa}\phi$, defined in (4),
should not diverge on the horizon. However, we can escape the no-hair theorem
Bravo-Gaete:2013dca ; Rinaldi:2012vy , for this we need to impose that the
conserved current’s radial component disappears in an identical way without
restricting the radial dependence of the scalar field Bravo-Gaete:2013dca :

$\alpha g_{rr}-\gamma G_{rr}=0.$ (6)

As $\phi^{\prime}_{\kappa}(r)\equiv\psi_{\kappa}(r)$, we can easily see that
this condition wipes out $\psi^{2}_{\kappa}(r)$, regardless of its behavior on
the horizon. The metric function $f(r)$ can be found using the equation (6).
It can be shown that the equation (4) is satisfied by the following solution

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle\frac{\alpha
L^{2}}{\gamma(2z+1)}-\left(\frac{r_{+}}{r}\right)^{2z+1},$ (7)
$\displaystyle\psi^{2}_{\kappa}(r)$ $\displaystyle=$
$\displaystyle-\frac{2L^{2}k(\alpha+\gamma\Lambda)}{\alpha\gamma
r^{2}}\frac{1}{f(r)\Theta_{\kappa}(r)},$ (8) $\displaystyle\Theta_{\kappa}(r)$
$\displaystyle=$ $\displaystyle
1+\kappa^{2}\left(\frac{r}{r_{+}}\right)^{\lambda};\quad\lambda=(z-1)\frac{\alpha-\gamma\Lambda}{\alpha+\gamma\Lambda}.$
(9)

Here $\phi^{{}^{\prime}}_{\kappa}\equiv\psi_{\kappa}$ define as a new field.
The Einstei9n-Horndeski field equations (3) and (4) are satisfied by the
equations (7) and (8) through the equation (9) for any value of the critical
exponent. As discussed by Anabalon:2013oea , we have an asymptotically dS or
AdS for $\alpha/\gamma<0$ or $\alpha/\gamma>0$, respectively. Thus, stability
requires that $(\alpha+\gamma\Lambda)$ is not negative, this leads to an
interval of the form $-\infty<\gamma\leq\alpha/(-\Lambda)$ Jiang:2017imk .
Here we consider $\Lambda=-(1+2z)/L^{2}$ with the following transformations in
the metric (5) with solution (7)-(8)

$\displaystyle
f(r)\to-\frac{\alpha}{\gamma\Lambda}{f}(r),r_{+}^{2z+1}\to-\frac{\alpha}{\gamma\Lambda}\;r_{+}^{2z+1},$
$\displaystyle
L\to\left(-\frac{{\alpha}}{\gamma\Lambda}\right)^{1/2}L,t\to-\frac{\gamma\Lambda}{\alpha}t,$
$\displaystyle
x\to\left(-\frac{{\gamma\Lambda}}{\alpha}\right)^{1/2}x,y\to\left(-\frac{{\gamma\Lambda}}{\alpha}\right)^{1/2}y,$
(10)

in order to put the black hole solution in the standard form

$\displaystyle{f}(r)$ $\displaystyle=$ $\displaystyle
1-\left(\frac{r_{+}}{r}\right)^{2z+1},$ (11) $\displaystyle\psi^{2}(r)$
$\displaystyle=$ $\displaystyle-\frac{\xi}{r^{2}f(r)\Theta_{\kappa}(r)}.$ (12)
$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{2\kappa(1+2z)(\alpha+\gamma\Lambda)}{\alpha^{2}}$ (13)

In fact, we have that the appearance of a black hole, which has a flat horizon
$\Re^{2}$, leads us to a physics in IR that corresponds to placing the
invariant theory of scale at a finite temperature Hartnoll:2009sz .

We have that for $f\to 1$ with $r\to 0$ it is a spacetime asymptotically AdS.
Thus, the further we advance in spacetime, entering the infrared (IR) region
for large values of $r$. However, in $r=r_{+}$, we have both $g_{tt}(r)$ and
$\psi^{2}_{\kappa}(r)$, the latter in action (1) ensures that this is a
genuine vacuum solution with $\Theta_{\kappa}(r)=$constant. Furthermore, we
have that the surface located at $r=r_{+}$ is infinitely shifted to red in
relation to an asymptotic observer. On the other hand, we can see by looking
at the equation (8), which $(\alpha+\gamma\Lambda)>0$, indicates ghost
freedom.

So, we have that the temperature of the black hole is given by

$\displaystyle T=\frac{-g^{\prime}_{tt}(r\rightarrow
r_{+})}{4\pi(\sqrt{-g_{tt}(r\rightarrow r_{+})g_{rr}(r\rightarrow r_{+})})},$
(14) $\displaystyle T=\frac{(1+2z)r^{z}_{+}}{4\pi}.$ (15)

##  IV The canonical Ensemble

Now, we present that free energy can be obtained by evaluating Euclidean
action as discussed by Hartnoll:2009sz ; Santos:2021orr . In fact, as already
well established, the path integral must include a Gibbons-Hawking boundary
term to provide the correct variational problem to the Horndeski gravity
Santos:2021orr . Thus, for the action (1), we have euclidean action:

$\displaystyle
I_{E}=-\kappa\int{\sqrt{g}d^{4}x\mathcal{L}_{H}}-2\kappa\int_{r\to\infty}{d^{3}x\sqrt{\bar{\gamma}}\mathcal{L}_{b}}+2\kappa\int_{r\to\infty}{d^{3}x\sqrt{\bar{\gamma}}\mathcal{L}_{ct}},$
(16)
$\displaystyle\mathcal{L}_{H}=(R-2\Lambda)+\frac{\gamma}{2}G_{\mu\nu}\Theta_{\kappa}(r)\nabla^{\mu}\phi\nabla^{\nu}\phi$
(17)
$\displaystyle\mathcal{L}_{b}=K^{({\bar{\gamma}})}-\Sigma^{(\bar{\gamma})}+\frac{\gamma}{4}\Theta_{\kappa}(r)(\nabla_{\mu}\phi\nabla_{\nu}\phi
n^{\mu}n^{\nu}-(\nabla\phi)^{2})K^{(\bar{\gamma})}+\frac{\gamma}{4}\Theta_{\kappa}(r)\nabla^{\mu}\phi\nabla^{\nu}\phi
K^{(\bar{\gamma})}_{\mu\nu}$ (18) $\displaystyle{\cal
L}_{ct}=c_{0}+c_{1}R+c_{2}R^{ij}R_{ij}+c_{3}R^{2}+b_{1}\Theta_{\kappa}(r)(\partial_{i}\phi\partial^{i}\phi)^{2}+...$
(19)

Here ${\cal L}_{ct}$ are boundary counterterms, they do not affect the bulk
dynamics and will be neglect. $\mathcal{L}_{b}$ corresponds to the Gibbons-
Hawking $\gamma$-dependent terms associated with the Horndeski gravity and
$n^{\mu}$ is an outward pointing unit normal vector to the boundary and
$K^{(\bar{\gamma})}=\bar{\gamma}^{\mu\nu}K^{({\bar{\gamma}})}_{\mu\nu}$ is the
trace of the extrinsic curvature. Beyond, we have that $\bar{\gamma}_{\mu\nu}$
and $\Sigma^{(\bar{\gamma})}$ are the induced metric and the surface tension
on the boundary $z\to 0$ Santos:2021orr . Thus, we have the Euclidean action
as

$\displaystyle I_{E}=-\frac{V_{2}\Sigma^{\kappa}_{z}L^{2}r^{z+1}_{+}}{8G_{N}}$
(20)
$\displaystyle\Sigma^{\kappa}_{z}=3+\frac{2(2+z^{2})}{z+2}-\frac{\gamma\xi(z+1)}{2+z}\,_{2}F_{1}\left(1,\frac{z+2}{\lambda};\frac{z+2}{\lambda}+1;-\kappa^{2}\right)$
(21)

Here $\Sigma^{\kappa}_{z}$ is a constant that includes all parameters of our
model. Now, we can work with the Canonical Ensemble where the free energy is
given by $F=-T\ln(Z)$, $Z$ is the partition function defined by $Z=e^{-I_{E}}$
where $I_{E}$ is the Euclidean action. Through the relation $\ln(Z)=-I_{E}$,
we have $F=TI_{E}$ Santos:2021orr . Thus, we can write the following
thermodynamic relation:

$\displaystyle\ln(Z)$ $\displaystyle=$
$\displaystyle-\sigma\Sigma^{\kappa}_{z}\left(\frac{4\pi}{(1+2z)}\right)^{1+3/z}T^{1+3/z},$
(22) $\displaystyle F$ $\displaystyle=$
$\displaystyle-\sigma\Sigma^{\kappa}_{z}\left(\frac{4\pi}{(1+2z)}\right)^{1+3/z}T^{2+3/z}$
(23)

Where $\sigma\equiv\frac{V_{2}L^{2}}{8G_{N}}$ and $M_{p}=G^{-1/2}$ is the mass
of Planck in units in which $\hbar=c=1$ Hawking:1982dh . Now, we can construct
the thermal states in the anti-de Sitter space, periodically identifying the
coordinate of the imaginary time $\tau$ with the period $\beta=1/T$. Thus, we
can find the expected value of energy, entropy and specific heat as

$\displaystyle\langle E\rangle$ $\displaystyle=$
$\displaystyle-\frac{\partial\ln(Z)}{\partial\beta}=\sigma\Sigma^{\kappa}_{z}(1+3/z)\left(\frac{4\pi}{(1+2z)}\right)^{1+3/z}T^{2+3/z},$
(24) $\displaystyle S$ $\displaystyle=$ $\displaystyle-\frac{\partial
F}{\partial
T}=\sigma\Sigma^{\kappa}_{z}(2+3/z)\left(\frac{4\pi}{(1+2z)}\right)^{1+3/z}T^{1+3/z},$
(25) $\displaystyle c$ $\displaystyle=$ $\displaystyle T\frac{\partial
S}{\partial
T}=\sigma\Sigma^{\kappa}_{z}(2+3/z)(1+3/z)\left(\frac{4\pi}{(1+2z)}\right)^{1+3/z}T^{1+3/z}.$
(26)

The free energy of the black hole, its entropy and internal energy satisfy the
thermodynamic relation $F=\langle E\rangle-TS$. Considering the of critical
exponent $z=3/2$, we have $\langle E\rangle\sim T^{4}$ that is similar to the
case of energy of the thermal radiation in Hawking:1982dh .

Figure 1: The figure shows the behavior of the equations (23), (24), (25) and
(26) considering the values for $\Lambda=-1$, $\alpha=-0.2$ with
$\gamma=-0.1$-$z=1/2$-$\kappa=1$ (solid), $\gamma=-0.2$-$z=1/3$-$\kappa=2$
(dashed), $\gamma=-0.3$-$z=2/5$-$\kappa=3$ (dot dashed), and
$\gamma=-0.4$-$z=4/5$-$\kappa=4$ (thick), respectively.

According to Fig.$\sim$1, we have that the values of $\langle E\rangle>0$ and
$c>0$ are in according to Hawking:1982dh  indicating a higher mass value of
the black hole, which is at least locally stable. In addition, the free energy
$F<0$ indicates that the system has global stability Rostami:2019xrx , but
this free energy provides $c>0$. In fact, large black holes (LBH) have
negative free energy. Beyond this thermodynamics quantities we can extract the
speed of sound and trace of the energy momentum tensor according to Ballon-
Bayona:2020xls ; Chen:2020ath  by mean the relation from thermodynamic
potential. Thus, we have the speed of sound as:

$c^{2}_{s}=\frac{S}{c}=\frac{z}{z+3},$ (27)

where in the usual case $z=3/2$, we have $c^{2}_{s}=1/3$, which is associated
with the conformal system Ballon-Bayona:2020xls ; Chen:2020ath . Now, we
present the trace of the energy momentum tensor:

$\displaystyle\langle T^{a}_{\ \ a}\rangle=\epsilon-3p=4F+TS,$ (28)
$\displaystyle\langle T^{a}_{\ \
a}\rangle=\sigma\Sigma^{\kappa}_{z}(2+3/z)\left(\frac{4\pi}{(1+2z)}\right)^{1+3/z}T^{2+3/z}-4\sigma\Sigma^{\kappa}_{z}\left(\frac{4\pi}{(1+2z)}\right)^{1+3/z}T^{2+3/z}$
(29)

where the scaled trace of the energy momentum tensor
($\langle\mathfrak{T}^{a}_{\ \ a}\rangle\equiv\langle T^{a}_{\ \
a}\rangle/T^{4}$) as a function of the temperature.

Figure 2: The figure shows the behavior of the trace of the energy momentum
tensor considering the values for $\Lambda=-1$, $\alpha=-0.2$ with
$\gamma=-0.1$-$z=1/2$-$\kappa=1$ (solid), $\gamma=-0.2$-$z=1/3$-$\kappa=2$
(dashed), $\gamma=-0.3$-$z=2/5$-$\kappa=3$ (dot dashed), and
$\gamma=-0.4$-$z=4/5$-$\kappa=4$ (thick), respectively.

Now, analyzing the equation (29), we have that for the value of $z=3$, implies
$\langle\mathfrak{T}^{a}_{\ \ a}\rangle\sim 1/T$ and for the high-temperature
regime, despite the influence of $\kappa$-Horndeski gravity,
$\langle\mathfrak{T}^{a}_{\ \ a}\rangle\to 0$, only for this value of critical
exponent, we have a restoration of the conformal symmetry. In this case, we
can scaled the thermodynamics quantities ${\cal F}\equiv F/T^{4}$,
$\langle{\cal E}\rangle\equiv=\langle E\rangle/T^{4}$, ${\cal S}\equiv
S/T^{4}$ and taking the value of $z=3$, we have such this increases as inverse
temperature ${\cal F}\sim 1/T$, $\langle{\cal E}\rangle\sim 1/T$, ${\cal
S}\sim 1/T$, respectively. In higher temperature, we have ${\cal F}\to 0$,
$\langle{\cal E}\rangle\to 0$ and ${\cal S}\to 0$ where the all quantities
decreases.

##  V Conclusion

In four dimensions, we show that black brane with asymptotically Lifshitz
spacetime for the any value of critical exponent is a solution of
$\kappa$-Horndeski’s theory. In fact, as we show the $\kappa$-algebra was
implement in the scalar field through the kinetic coupling and this auxiliary
equation (9), was crucial to satisfy the equations of motion to any value of
critical exponent in these new formalism without a special truncation of the
critical exponent in the theory. In the canonical Ensemble, we show that the
free energy, entropy and internal energy satisfy the thermodynamic
relationship $F=\langle E\rangle-TS$. Thus, together with specific heat, it
plays an important role in the investigation of thermal stability. As the free
energy is negative $F<0$ we have global stability, as shown in Fig.$\sim$1
with $c>0$, showing that the system has local stability Hawking:1982dh .

However, as the energy $\langle E\rangle$ increasing in Fig.$\sim$1, this
means that the black hole "captured" a new mass energy ( matter or radiation
goes beyond the surface surrounding the black hole, which we call the
horizon), where the sector of matter is given only by the kinetic term
associated with the scalar field. On the other hand, if the energy
$\langle{\cal E}\rangle\to 0$ decreases to $z=3$, it suggests that the only
mechanism proposed to decrease the mass of a black hole is the evaporation of
Hawking radiation. That is a process of quantum origin and arises from the
creation of pairs of particles from the vacuum energy in the horizon region.

We would like to thank CNPq and CAPES for partial financial support. I would
also like to thank Mokhtar Hassine for fruitful discussions at the end of this
work.

## References

  * (1) L. Heisenberg, A systematic approach to generalisations of General Relativity and their cosmological implications, Phys. Rept. 796, 1 (2019), [arXiv:1807.01725 [gr-qc]].
  * (2) G. W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. 10, 363 (1974).
  * (3) J. P. Bruneton, M. Rinaldi, A. Kanfon, A. Hees, S. Schlogel and A. Fuzfa, Fab Four: When John and George play gravitation and cosmology, Adv. Astron. 2012, 430694 (2012), [arXiv:1203.4446 [gr-qc]].
  * (4) F. A. Brito and F. F. Dos Santos, Braneworlds in Horndeski gravity, arXiv:1810.08196 [hep-th].
  * (5) A. Cisterna and C. Erices, Asymptotically locally AdS and flat black holes in the presence of an electric field in the Horndeski scenario, Phys. Rev. D 89, 084038 (2014), [arXiv:1401.4479 [gr-qc]].
  * (6) A. Cisterna, T. Delsate, L. Ducobu and M. Rinaldi, Slowly rotating neutron stars in the nonminimal derivative coupling sector of Horndeski gravity, Phys. Rev. D 93, no. 8, 084046 (2016), [arXiv:1602.06939 [gr-qc]].
  * (7) G. W. Gibbons, Anti-de-Sitter spacetime and its uses, arXiv:1110.1206 [hep-th].
  * (8) J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38, 1113 (1999) [Adv. Theor. Math. Phys. 2, 231 (1998)], [hep-th/9711200].
  * (9) S. A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26, 224002 (2009), [arXiv:0903.3246 [hep-th]].
  * (10) S. Sachdev, What can gauge-gravity duality teach us about condensed matter physics?, Ann. Rev. Condensed Matter Phys. 3, 9 (2012), [arXiv:1108.1197 [cond-mat.str-el]].
  * (11) P. Kovtun, D. T. Son and A. O. Starinets, Holography and hydrodynamics: Diffusion on stretched horizons, JHEP 0310, 064 (2003), [hep-th/0309213].
  * (12) P. Kovtun, D. T. Son and A. O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94, 111601 (2005), [hep-th/0405231].
  * (13) X. H. Feng, H. S. Liu, H. Lü and C. N. Pope, Black Hole Entropy and Viscosity Bound in Horndeski Gravity, JHEP 1511, 176 (2015), [arXiv:1509.07142 [hep-th]].
  * (14) F. A. Brito and F. F. Santos, Black branes in asymptotically Lifshitz spacetime and viscosity/entropy ratios in Horndeski gravity, arXiv:1901.06770 [hep-th].
  * (15) F. F. Santos, it Aplicações do Setor John da Gravidade de Horndeski nos Cenários de Brana Negra e Relação de viscosidade/entropia, Mundo Brana e Cosmologia (In Portuguese), [arXiv:2006.06550 [hep-th]].
  * (16) M. Sadeghi, Black Brane Solution in Rastall AdS Massive Gravity and Viscosity Bound, Mod. Phys. Lett. A 33, no. 37, 1850220 (2018), [arXiv:1809.08698 [hep-th]].
  * (17) V. Oikonomou and F. Fronimos, Reviving non-Minimal Horndeski-like Theories after GW170817: Kinetic Coupling Corrected Einstein-Gauss-Bonnet Inflation, [arXiv:2006.05512 [gr-qc]].
  * (18) C. Charmousis, E. J. Copeland, A. Padilla and P. M. Saffin, General second order scalar-tensor theory, self tuning, and the Fab Four, Phys. Rev. Lett. 108, 051101 (2012), [arXiv:1106.2000 [hep-th]].
  * (19) C. Charmousis, E. J. Copeland, A. Padilla and P. M. Saffin, Self-tuning and the derivation of a class of scalar-tensor theories, Phys. Rev. D 85, 104040 (2012), [arXiv:1112.4866 [hep-th]].
  * (20) M. Bravo-Gaete and M. Hassaine, Lifshitz black holes with a time-dependent scalar field in a Horndeski theory, Phys. Rev. D 89, 104028 (2014), [arXiv:1312.7736 [hep-th]].
  * (21) A. Anabalon, A. Cisterna and J. Oliva, Asymptotically locally AdS and flat black holes in Horndeski theory, Phys. Rev. D 89, 084050 (2014), [arXiv:1312.3597 [gr-qc]].
  * (22) M. Rinaldi, Black holes with non-minimal derivative coupling, Phys. Rev. D 86, 084048 (2012), [arXiv:1208.0103 [gr-qc]].
  * (23) C. Deffayet, X. Gao, D. A. Steer and G. Zahariade, From k-essence to generalised Galileons, Phys. Rev. D 84, 064039 (2011), [arXiv:1103.3260 [hep-th]].
  * (24) M. Rinaldi, Mimicking dark matter in Horndeski gravity, Phys. Dark Univ. 16, 14 (2017), [arXiv:1608.03839 [gr-qc]].
  * (25) A. Cisterna, M. Hassaine, J. Oliva and M. Rinaldi, Axionic black branes in the k-essence sector of the Horndeski model, Phys. Rev. D 96, no. 12, 124033 (2017), [arXiv:1708.07194 [hep-th]].
  * (26) J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, Essential Building Blocks of Dark Energy, JCAP 1308, 025 (2013), [arXiv:1304.4840 [hep-th]].
  * (27) M. Zumalacárregui and J. García-Bellido, Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian, Phys. Rev. D 89, 064046 (2014), [arXiv:1308.4685 [gr-qc]].
  * (28) A. A. Starobinsky, S. V. Sushkov and M. S. Volkov, The screening Horndeski cosmologies, JCAP 1606, no. 06, 007 (2016), [arXiv:1604.06085 [hep-th]].
  * (29) M. Rostami, J. Sadeghi, S. Miraboutalebi, B. Pourhassan and A. A. Masoudi, Phase transition of modified Horndeski gravity with new method, arXiv:1909.01799 [gr-qc].
  * (30) T. Nakama and J. Yokoyama, Micro black holes formed in the early Universe and their cosmological implications, Phys. Rev. D 99, no. 6, 061303 (2019), [arXiv:1811.05049 [gr-qc]].
  * (31) S. Dimopoulos and G. L. Landsberg, Black holes at the LHC, Phys. Rev. Lett. 87, 161602 (2001), [hep-ph/0106295].
  * (32) S. W. Hawking and D. N. Page, Thermodynamics of Black Holes in anti-De Sitter Space, Commun. Math. Phys. 87, 577 (1983).
  * (33) Y. C. Ong, Hawking Evaporation Time Scale of Topological Black Holes in Anti-de Sitter Spacetime, Nucl. Phys. B 903, 387 (2016), [arXiv:1507.07845 [gr-qc]].
  * (34) O. Baake, M. Bravo-Gaete and M. Hassaine, Spinning black holes for generalized scalar tensor theories in three dimensions, [arXiv:2005.10869 [hep-th]].
  * (35) L. Susskind and J. Lindesay, An introduction to black holes, information and the string theory revolution: The holographic universe,
  * (36) F. F. Santos, E. F. Capossoli and H. Boschi-Filho, AdS/BCFT correspondence and BTZ black hole thermodynamics within Horndeski gravity, Phys. Rev. D 104, no.6, 066014 (2021), [arXiv:2105.03802 [hep-th]].
  * (37) Y. Z. Li, H. Lu and H. Y. Zhang, Scale Invariance vs. Conformal Invariance: Holographic Two-Point Functions in Horndeski Gravity, Eur. Phys. J. C 79, no. 7, 592 (2019), [arXiv:1812.05123 [hep-th]].
  * (38) C. Tsallis, Possible Generalization of Boltzmann-Gibbs Statistics, J. Statist. Phys. 52, 479 (1988).
  * (39) B. G. da Costa, I. S. Gomez and M. Portesi, $\kappa$-Deformed quantum and classical mechanics for a system with position-dependent effective mass, J. Math. Phys. 61, no.8, 082105 (2020), [arXiv:2007.11184 [quant-ph]].
  * (40) Ernesto P. Borges A possible deformed algebra and calculus inspired in nonextensive thermostatistics, Physica A, (2003),
  * (41) W. J. Jiang, H. S. Liu, H. Lu and C. N. Pope, DC Conductivities with Momentum Dissipation in Horndeski Theories, JHEP 1707, 084 (2017), [arXiv:1703.00922 [hep-th]].
  * (42) D. N. Page, Finite upper bound for the Hawking decay time of an arbitrarily large black hole in anti–de Sitter spacetime, Phys. Rev. D 97, no. 2, 024004 (2018), [arXiv:1507.02682 [hep-th]].
  * (43) H. Xu and M. H. Yung, Black hole evaporation in Lovelock gravity with diverse dimensions, Phys. Lett. B 794, 77 (2019), [arXiv:1904.06503 [gr-qc]].
  * (44) H. Xu and M. H. Yung, Black hole evaporation in Conformal (Weyl) gravity, Phys. Lett. B 793, 97 (2019), [arXiv:1811.07309 [gr-qc]].
  * (45) Y. C. Ong, The Persistence of the Large Volumes in Black Holes, Gen. Rel. Grav. 47, no. 8, 88 (2015), [arXiv:1503.08245 [gr-qc]].
  * (46) N. K. A., C. L. A. Rizwan, A. K. M. and M. S. Ali, Photon Orbits and Thermodynamic Phase Transition of Regular AdS Black Holes, arXiv:1912.11909 [gr-qc].
  * (47) A. Ballon-Bayona, H. Boschi-Filho, E. F. Capossoli and D. M. Rodrigues, Criticality from Einstein-Maxwell-dilaton holography at finite temperature and density,” Phys. Rev. D 102, no. 12, 126003 (2020), [arXiv:2006.08810 [hep-th]].
  * (48) X. Chen, L. Zhang, D. Li, D. Hou and M. Huang, Gluodynamics and deconfinement phase transition under rotation from holography,” arXiv:2010.14478 [hep-ph].