# This is the title

PAULSEN AND PROJECTION PROBLEMS FOR BANACH SPACES
K. MAHESH KRISHNA

Post Doctoral Fellow

Statistics and Mathematics Unit

Indian Statistical Institute, Bangalore Centre

Karnataka 560 059 India

Email<EMAIL_ADDRESS>
Date:

Abstract: Based on the two decades old celebrated Paulsen problem and its
solutions for Hilbert spaces by Kwok, Lau, Lee, Ramachandran, Hamilton, and
Moitra, we formulate Paulsen problem for Banach spaces. We also formulate
projection problem for Banach spaces.

Keywords: Paulsen problem, Projection problem, Approximate Schauder Frame.

Mathematics Subject Classification (2020): 42C15.

###### Contents

  1. 1 Introduction
  2. 2 Paulsen and projection problems for Banach spaces

##  1\. Introduction

Infinite dimensional frame theory, originated from the work of Duffin and
Schaeffer [28, 23] influenced the development of frame theory for finite
dimensional Hilbert spaces towards the end of 20th century and theory
developed rapidly in the first decade of this century. Revolutionary works on
finite frames are [12, 19, 9, 7, 1, 16, 3, 11, 30, 5, 43, 29, 44, 20]. Finite
dimensional definition of frames reads as follows. Letter $\mathcal{H}$ always
denotes a finite dimensional Hilbert space. We denote the identity operator on
$\mathcal{H}$ by $I_{\mathcal{H}}.$

######  Definition 1.1.

[35, 3] A collection $\\{\tau_{j}\\}_{j=1}^{n}$ in a finite dimensional
Hilbert space $\mathcal{H}$ over $\mathbb{K}$ ($\mathbb{R}$ or $\mathbb{C}$)
is said to be a frame for $\mathcal{H}$ if there exist $a,b>0$ such that

$\displaystyle a\|h\|^{2}\leq\sum_{j=1}^{n}|\langle h,\tau_{j}\rangle|^{2}\leq
b\|h\|^{2},\quad\forall h\in\mathcal{H}.$

A frame $\\{\tau_{j}\\}_{j=1}^{n}$ for $\mathcal{H}$ is said to be Parseval if

$\displaystyle h=\sum_{j=1}^{n}\langle h,\tau_{j}\rangle\tau_{j},\quad\forall
h\in\mathcal{H}.$

A frame $\\{\tau_{j}\\}_{j=1}^{n}$ for $\mathcal{H}$ is said to be tight if

$\displaystyle a\|h\|^{2}=\sum_{j=1}^{n}|\langle
h,\tau_{j}\rangle|^{2},\quad\forall h\in\mathcal{H}.$

A frame $\\{\tau_{j}\\}_{j=1}^{n}$ for $\mathcal{H}$ is said to be unit norm
if $\|\tau_{j}\|=1$, for all $1\leq j\leq n$.

A frame $\\{\tau_{j}\\}_{j=1}^{n}$ for $\mathcal{H}$ induces three operators
defined as follows.

  1. (i)

Analysis operator $\theta_{\tau}:\mathcal{H}\ni
h\mapsto\theta_{\tau}h\coloneqq(\langle
h,\tau_{j}\rangle)_{j=1}^{n}\in\mathbb{K}^{n}$.

  2. (ii)

Synthesis operator
$\theta_{\tau}^{*}:\mathbb{K}^{n}\ni(a_{j})_{j=1}^{n}\mapsto\theta_{\tau}^{*}(a_{j})_{j=1}^{n}\coloneqq\sum_{j=1}^{n}a_{j}\tau_{j}\in\mathcal{H}$.

  3. (iii)

Frame operator $S_{\tau}:\mathcal{H}\ni h\mapsto
S_{\tau}h\coloneqq\sum_{j=1}^{n}\langle
h,\tau_{j}\rangle\tau_{j}\in\mathcal{H}$.

Frame condition then gives that the analysis operator is injective, synthesis
operator is surjective and the frame operator is positive and invertible.
Further, the frame operator factors as
$S_{\tau}=\theta_{\tau}^{*}\theta_{\tau}$. Since frame operator is positive
and invertible, one has the conclusion that
$\\{S_{\tau}^{-1/2}\tau_{j}\\}_{j=1}^{n}$ is a Parseval frame for
$\mathcal{H}$. In general, given a frame, construction of Parseval frame in
this way is difficult. Also, given a collection it is easily to verify that
whether it is a frame than to verify that it is a Parseval frame. This leads
to the following notions.

######  Definition 1.2.

[6] A Parseval frame $\\{\tau_{j}\\}_{j=1}^{n}$ for a $d$-dimensional Hilbert
space $\mathcal{H}$ is called an equal norm Parseval frame if

$\displaystyle\|\tau_{j}\|^{2}=\frac{d}{n},\quad\forall 1\leq j\leq n.$

######  Definition 1.3.

[6] A frame $\\{\tau_{j}\\}_{j=1}^{n}$ for a $d$-dimensional Hilbert space
$\mathcal{H}$ is called an $\varepsilon$-nearly Parseval frame
($\varepsilon<1$) if

$\displaystyle(1-\varepsilon)I_{\mathcal{H}}\leq
S_{\tau}\leq(1+\varepsilon)I_{\mathcal{H}}.$

######  Definition 1.4.

[6] A frame $\\{\tau_{j}\\}_{j=1}^{n}$ for a $d$-dimensional Hilbert space
$\mathcal{H}$ is called an $\varepsilon$-nearly equal norm frame
($\varepsilon<1$) if

$\displaystyle(1-\varepsilon)\frac{d}{n}\leq\|\tau_{j}\|^{2}\leq(1+\varepsilon)\frac{d}{n},\quad\forall
1\leq j\leq n.$

######  Definition 1.5.

[6] A frame for a $d$-dimensional Hilbert space $\mathcal{H}$ which is both
$\varepsilon$-nearly equal norm and $\varepsilon$-nearly Parseval is called as
$\varepsilon$-nearly equal norm Parseval frame.

Distance between two collections in Hilbert space are measured using following
distance.

######  Definition 1.6.

[6] Distance between two collections $\\{\tau_{j}\\}_{j=1}^{n}$,
$\\{\omega_{j}\\}_{j=1}^{n}$ in a Hilbert space $\mathcal{H}$ is defined as

$\displaystyle\operatorname{dist}(\\{\tau_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n})\coloneqq\left(\sum_{j=1}^{n}\|\tau_{j}-\omega_{j}\|^{2}\right)^{\frac{1}{2}}.$

We then have the fundamental Paulsen problem which is originated from the work
of Holmes and Paulsen [36]. Basic motivation is the following three results.

######  Theorem 1.7.

[36] There is an algorithm for turning every frame into an equal norm frame
with same frame operator.

######  Theorem 1.8.

[9] Let $\\{\tau_{j}\\}_{j=1}^{n}$ be an $\varepsilon$-nearly equal norm frame
for a $d$-dimensional Hilbert space $\mathcal{H}$. Then the closest equal norm
frame to $\\{\tau_{j}\\}_{j=1}^{n}$ is given by $\\{\omega_{j}\\}_{j=1}^{n}$,
where

$\displaystyle\omega_{j}\coloneqq\left(\frac{\sum_{k=1}^{n}\|\tau_{k}\|}{n}\right)\frac{\tau_{j}}{\|\tau_{j}\|},\quad\forall
1\leq j\leq n.$

######  Theorem 1.9.

[4, 17] Let $\\{\tau_{j}\\}_{j=1}^{n}$ be a frame for a $d$-dimensional
Hilbert space $\mathcal{H}$. Then $\\{S_{\tau}^{-1/2}\tau_{j}\\}_{j=1}^{n}$ is
the closest Parseval frame to $\\{\tau_{j}\\}_{j=1}^{n}$, i.e.,

$\displaystyle\sum_{j=1}^{n}\left\|S_{\tau}^{\frac{-1}{2}}\tau_{j}-\tau_{j}\right\|^{2}=\inf\left\\{\sum_{j=1}^{n}\left\|\tau_{j}-\omega_{j}\right\|^{2}:\\{\omega_{j}\\}_{j=1}^{n}\text{
is a Parseval frame for }\mathcal{H}\right\\}.$

Further, $\\{S_{\tau}^{-1/2}\tau_{j}\\}_{j=1}^{n}$ is the unique minimizer.
Moreover, if $\\{\tau_{j}\\}_{j=1}^{n}$ is any $\varepsilon$-nearly Parseval
frame for $\mathcal{H}$, then

$\displaystyle\sum_{j=1}^{n}\left\|S_{\tau}^{\frac{-1}{2}}\tau_{j}-\tau_{j}\right\|^{2}\leq
d(2-\varepsilon-2\sqrt{1-\varepsilon})\leq\frac{d\varepsilon^{2}}{4}.$

######  Problem 1.10.

[6] (Paulsen problem) Find the function
$f:(0,1)\times\mathbb{N}\times\mathbb{N}\to[0,\infty)$ so that for any
$\varepsilon$-nearly equal norm Parseval frame $\\{\tau_{j}\\}_{j=1}^{n}$ for
d-dimensional Hilbert space $\mathcal{H}$, there is an equal norm Parseval
frame $\\{\omega_{j}\\}_{j=1}^{n}$ for $\mathcal{H}$ satisfying

$\displaystyle\operatorname{dist}^{2}(\\{\tau_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n})=\sum_{j=1}^{n}\|\tau_{j}-\omega_{j}\|^{2}\leq
f(\varepsilon,n,d).$

Moreover, whether $f$ depends on $n$?

It is clear that Problem 1.10 can also be stated as follows.

######  Problem 1.11.

[6](Paulsen problem) Find the function
$f:(0,1)\times\mathbb{N}\times\mathbb{N}\to[0,\infty)$ so that for any
$\varepsilon$-nearly equal norm Parseval frame $\\{\tau_{j}\\}_{j=1}^{n}$ for
d-dimensional Hilbert space $\mathcal{H}$,

$\displaystyle\inf\\{\operatorname{dist}^{2}(\\{\tau_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n}):\\{\omega_{j}\\}_{j=1}^{n}\text{
is an equal norm Parseval frame for }\mathcal{H}\\}\leq f(\varepsilon,n,d).$

Moreover, whether $f$ depends on $n$?

The function $f$ in Definition 1.10 and in Definition 1.11 is known as Paulsen
function. One first asks whether such a function exists. Using compactness of
the unit sphere in finite dimensional Hilbert space, Hadwin proved the
following.

######  Theorem 1.12.

[4] Solution $f$ to Paulsen problem exists.

Casazza [10] noticed that Paulsen function is bounded below which is
independent of the number of elements in the frame.

######  Proposition 1.13.

[10] Paulsen function $f$ satisfies

$\displaystyle
f(\varepsilon,n,d)\geq\varepsilon^{2}d,\quad\forall\varepsilon>0,\forall
d\in\mathbb{N}.$

Using the system of ordinary differential equations and using the notion of
frame energy, Bodmann and Casazza made the first breakthrough for Paulsen
problem in 2010 [4].

######  Theorem 1.14.

[4] Let $n$ and $d$ be relatively prime and let $\varepsilon<1/2$. If
$\\{\tau_{j}\\}_{j=1}^{n}$ is any $\varepsilon$-nearly equal norm frame for a
$d$-dimensional Hilbert space $\mathcal{H}$, then there is an equal norm
Parseval frame $\\{\omega_{j}\\}_{j=1}^{n}$ for $\mathcal{H}$ such that

$\displaystyle\operatorname{dist}^{2}(\\{\tau_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n})\leq\frac{29}{8}d^{2}n(n-1)^{8}\varepsilon.$

Using the notion of Hilbert-Schmidt norm of operator, Casazza, Fickus, and
Mixon [13] gave the following version of Paulsen problem.

######  Problem 1.15.

[13] (Paulsen problem) Given $n,d\in\mathbb{N}$, find positive $\delta(n,d)$,
$c(n,d)$ and $\alpha(n,d)$ such that given any unit norm frame
$\\{\tau_{j}\\}_{j=1}^{n}$ for $d$-dimensional Hilbert space $\mathcal{H}$
satisfying

$\displaystyle\left\|S_{\tau}-\frac{n}{d}I_{\mathcal{H}}\right\|_{\operatorname{HS}}\leq\delta(n,d),$

there exists a unit norm tight frame $\\{\omega_{j}\\}_{j=1}^{n}$ for
$\mathcal{H}$ such that

$\displaystyle\|\theta_{\omega}^{*}-\theta_{\tau}^{*}\|_{\operatorname{HS}}\leq
c(n,d)\left\|S_{\tau}-\frac{n}{d}I_{\mathcal{H}}\right\|_{\operatorname{HS}}^{\alpha(n,d)},$

where HS denotes the Hilbert-Schmidt norm of the operator.

Using gradient of the frame potential, following particular cases of Problem
1.15 have been solved in 2012 [13].

######  Theorem 1.16.

[13] Let $n$ and $d$ be relatively prime and let $0<t<1/{2n}$. Let
$\\{\tau^{(0)}_{j}\\}_{j=1}^{n}$ be a unit norm tight frame for a
$d$-dimensional Hilbert space $\mathcal{H}$ such that

$\displaystyle\left\|S^{(0)}_{\tau}-\frac{n}{d}I_{\mathcal{H}}\right\|_{\operatorname{HS}}^{2}\leq\frac{2}{d^{3}}.$

Define $\\{\omega_{j}^{(k)}\\}_{j=1}^{n}$ as follows.

$\displaystyle\omega_{j}^{(k)}\coloneqq S_{\tau}^{(k)}\tau_{j}^{(k)}-\langle
S_{\tau}^{(k)}\tau_{j}^{(k)},\tau_{j}^{(k)}\rangle\tau_{j}^{(k)},\quad\forall
1\leq j\leq n,\forall k\geq 0.$

Now define $\\{\tau_{j}^{(k)}\\}_{j=1}^{n}$ as follows.

$\displaystyle\tau_{j}^{(k+1)}\coloneqq\begin{dcases}\cos(\|\omega_{j}^{(k)}\|t)\tau_{j}^{(k)}-\sin(\|\omega_{j}^{(k)}\|t)\frac{\omega_{j}^{(k)}}{\|\omega_{j}^{(k)}\|}&\text{if
}\omega_{j}^{(k)}\neq 0\\\ \tau_{j}^{(k)}&\text{if }\omega_{j}^{(k)}=0\\\
\end{dcases},\quad\forall 1\leq j\leq n,\forall k\geq 0.$

Then the limit of $\\{\tau_{j}^{(k)}\\}_{j=1}^{n}$ as $k\to\infty$ exists,
denoted by $\\{\tau_{j}^{(\infty)}\\}_{j=1}^{n}$ is a unit norm tight frame
for $\mathcal{H}$ and satisfies

$\displaystyle\left\|S^{(\infty)}_{\tau}-S^{(0)}_{\tau}\right\|_{\operatorname{HS}}\leq\frac{4d^{20}n^{8.5}}{1-2nt}\left\|S^{(0)}_{\tau}-\frac{n}{d}I_{\mathcal{H}}\right\|_{\operatorname{HS}}.$

######  Theorem 1.17.

[13] Let $\varepsilon\leq 1/{2n}$. If $\\{\tau_{j}\\}_{j=1}^{n}$ is any
$\varepsilon$-orthogonally partitionable unit norm frame for a $d$-dimensional
Hilbert space $\mathcal{H}$, then there is an orthogonally partitionable unit
norm frame $\\{\omega_{j}\\}_{j=1}^{n}$ for $\mathcal{H}$ such that

$\displaystyle\|\theta_{\omega}^{*}-\theta_{\tau}^{*}\|_{\operatorname{HS}}\leq\sqrt{2n}(\varepsilon
d)^{\frac{1}{3}}.$

######  Theorem 1.18.

[13] Let $n$ and $d$ be not relatively prime. If $\\{\tau_{j}\\}_{j=1}^{n}$ is
any unit norm frame for a $d$-dimensional Hilbert space $\mathcal{H}$, then
there is a unit norm frame $\\{\omega_{j}\\}_{j=1}^{n}$ for $\mathcal{H}$
which is either tight or is orthogonally partitionable with equal redundancies
in each of the two partitioned subsets such that

$\displaystyle\|\theta_{\omega}^{*}-\theta_{\tau}^{*}\|_{\operatorname{HS}}\leq
3d^{\frac{6}{7}}\sqrt{n}\left\|S_{\tau}-\frac{n}{d}I_{\mathcal{H}}\right\|_{\operatorname{HS}}^{\frac{1}{7}}.$

Projection problem is another classical problem in Hilbert space theory which
states as follows.

######  Problem 1.19.

[6] (Projection problem)  Let $\mathcal{H}$ be a d-dimensional Hilbert space
with orthonormal basis $\\{u_{k}\\}_{k=1}^{d}$. Find the function
$g:(0,1)\times\mathbb{N}\times\mathbb{N}\to[0,\infty)$ satisfying the
following: If $P:\mathcal{H}\to\mathcal{H}$ is an orthogonal projection of
rank $n$ satisfying

$\displaystyle(1-\varepsilon)\frac{n}{d}\leq\|Pu_{k}\|^{2}\leq(1+\varepsilon)\frac{n}{d},\quad\forall
1\leq k\leq d,$

then there exists an orthogonal projection $Q:\mathcal{H}\to\mathcal{H}$ with

$\displaystyle\|Qu_{k}\|^{2}=\frac{n}{d},\quad\forall 1\leq k\leq d,$

satisfying

$\displaystyle\sum_{k=1}^{d}\|Pu_{k}-Qu_{k}\|^{2}\leq g(\varepsilon,n,d).$

Using chordal distance between subspaces [24], Cahill and Casazza [6] showed
that Paulsen problem has a solution if and only if projection problem has
solution.

######  Theorem 1.20.

[6] If $f$ is the function for the Paulsen problem and $g$ is the function for
the projection problem, then

$\displaystyle g(\varepsilon,n,d)\leq 4f(\varepsilon,n,d)\leq
8f(\varepsilon,n,d),\quad\forall\varepsilon,n,d.$

Using Naimark complement of frames (see [14, 18, 21, 25]) Cahill and Casazza
[6] proved the following result.

######  Theorem 1.21.

[6] Let $n>d$. If $f$ is the function for the Paulsen problem, then

$\displaystyle f(\varepsilon,n,d)\leq 8f\left(\frac{d}{n-d},n,n-d\right).$

Theorem 1.21 then gives the following result.

######  Theorem 1.22.

[6] To solve the Paulsen problem for $d$-dimensional Hilbert space
$\mathcal{H}$, it suffices to solve it for Parseval frames
$\\{\tau_{j}\\}_{j=1}^{n}$ for $\mathcal{H}$ with $d\leq n\leq 2d$.

In 2017, using operator scaling algorithm and smoothed analysis, Kwok, Lau,
Lee, and Ramachandran [42, 41] resolved Paulsen problem by deriving the
following result.

######  Theorem 1.23.

[42, 41] For any $\varepsilon$-nearly equal norm Parseval frame
$\\{\tau_{j}\\}_{j=1}^{n}$ for $\mathbb{R}^{d}$, there is an equal norm
Parseval frame $\\{\omega_{j}\\}_{j=1}^{n}$ for $\mathbb{R}^{d}$ satisfying

$\displaystyle\operatorname{dist}^{2}(\\{\tau_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n})\leq
O(\varepsilon d^{\frac{13}{2}}).$

In other words, $f$ does not depend upon $n$.

In 2018, using radial isotropic position, Hamilton and Moitra [34, 33] gave
another proof of Paulsen problem which improved Theorem 1.23.

######  Theorem 1.24.

[34, 33] For any $\varepsilon$-nearly equal norm Parseval frame
$\\{\tau_{j}\\}_{j=1}^{n}$ for $\mathbb{R}^{d}$, there is an equal norm
Parseval frame $\\{\omega_{j}\\}_{j=1}^{n}$ for $\mathbb{R}^{d}$ satisfying

$\displaystyle\operatorname{dist}^{2}(\\{\tau_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n})\leq
20\varepsilon d^{2}.$

In other words, $f(\varepsilon,n,d)=20\varepsilon d^{2}$.

##  2\. Paulsen and projection problems for Banach spaces

Throughout this paper, we consider only finite dimensional Banach spaces. Let
$\mathcal{X}$ be a $d$-dimensional Banach space. Given an operator $T$ on
$\mathcal{X}$, by $\sigma(T)$, we mean the spectrum of $T$.

Motivated from the foundational work of Casazza, Han, and Larson [15] and from
the fundamental work of Daubechies and DeVore [26], Casazza, Dilworth, Odell,
Schlumprecht, and Zsak [8] introduced the notion of approximate Schauder
frames for Banach spaces which led to the notion of approximate Schauder
frames. Later, in the course of defining frame potential for Banach space and
from the notion of Auerbach basis [46, 32, 2, 27], Chavez-Domingues, Freeman,
and Kornelson [22] introduced the notion of unit norm tight frames for Banach
spaces.

######  Definition 2.1.

[31, 45, 15, 22, 40] Let $\\{\tau_{j}\\}_{j=1}^{n}$ be a collection in a
Banach space $\mathcal{X}$ and $\\{f_{j}\\}_{j=1}^{n}$ be a collection in
$\mathcal{X}^{*}$ (dual of $\mathcal{X}$). The pair
$(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ is said to be an
approximate Schauder frame (ASF) for $\mathcal{X}$ if the frame operator

$\displaystyle S_{f,\tau}:\mathcal{X}\ni x\mapsto
S_{f,\tau}x\coloneqq\sum_{j=1}^{n}f_{j}(x)\tau_{j}\in\mathcal{X}$

is invertible. Let $\lambda\in\mathbb{K}\setminus\\{0\\}$. An ASF
$(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ is said to be
$\lambda$-tight if $S_{f,\tau}=\lambda I_{\mathcal{X}}$, where
$I_{\mathcal{X}}$ is the identity operator on $\mathcal{X}$. If
$S_{f,\tau}=I_{\mathcal{X}}$, then we say
$(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ is a Parseval ASF or
framing or Schauder frame for $\mathcal{X}$. A tight ASF
$(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ for $\mathcal{X}$ is called
as a finite unit norm tight frame if

$\displaystyle\|f_{j}\|=\|\tau_{j}\|=f_{j}(\tau_{j})=1,\quad\forall 1\leq
j\leq n.$

We now formulate Banach space versions of Definitions 1.2, 1.3, 1.4 1.5 and
1.6.

######  Definition 2.2.

A Parseval frame $(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ for a
$d$-dimensional Banach space $\mathcal{X}$ is called an equal norm Parseval
ASF if

$\displaystyle\|\tau_{j}\|^{2}=\|f_{j}\|^{2}=f_{j}(\tau_{j})=\frac{d}{n},\quad\forall
1\leq j\leq n.$

######  Definition 2.3.

An ASF $(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ for a
$d$-dimensional Banach space $\mathcal{X}$ is called an $\varepsilon$-nearly
Parseval ASF ($\varepsilon<1$) if

$\displaystyle\sigma(S_{f,\tau})\subseteq(1-\varepsilon,1+\varepsilon).$

######  Definition 2.4.

An ASF $(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ for a
$d$-dimensional Banach space $\mathcal{X}$ is called an $\varepsilon$-nearly
equal norm ASF ($\varepsilon<1$) if

$\displaystyle(1-\varepsilon)\frac{d}{n}\leq\|\tau_{j}\|^{2}=f_{j}(\tau_{j})=\|f_{j}\|^{2}\leq(1+\varepsilon)\frac{d}{n},\quad\forall
1\leq j\leq n.$

######  Definition 2.5.

An ASF which is both $\varepsilon$-nearly equal norm and $\varepsilon$-nearly
Parseval is called as $\varepsilon$-nearly equal norm Parseval ASF.

######  Definition 2.6.

Let $\\{\tau_{j}\\}_{j=1}^{n}$, $\\{\omega_{j}\\}_{j=1}^{n}$ be collections in
a $d$-dimensional Banach space $\mathcal{X}$ and $\\{f_{j}\\}_{j=1}^{n}$,
$\\{g_{j}\\}_{j=1}^{n}$ be collections in $\mathcal{X}^{*}$. Distance between
two tuples $(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ and
$(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n})$ is defined as

(1)
$\displaystyle\operatorname{dist}((\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n}),(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n}))\coloneqq\left(\sum_{j=1}^{n}\frac{1}{2}(\|\tau_{j}-\omega_{j}\|^{2}+\|f_{j}-g_{j}\|^{2})\right)^{\frac{1}{2}}.$

######  Remark 2.7.

  1. (i)

Rather defining distance as in Definition (1), we would have also been defined
distance as

$\displaystyle\operatorname{dist}^{*}((\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n}),(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n}))\coloneqq\frac{1}{2}\left(\left(\sum_{j=1}^{n}\|\tau_{j}-\omega_{j}\|^{2}\right)^{\frac{1}{2}}+\left(\sum_{j=1}^{n}\|f_{j}-g_{j}\|^{2}\right)^{\frac{1}{2}}\right).$

However, we then have

$\displaystyle\operatorname{dist}^{*}((\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n}),(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n}))$
$\displaystyle\leq\operatorname{dist}((\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n}),(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n})),$
$\displaystyle\quad\forall(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n}),(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n}).$

  2. (ii)

We can also try by replacing the distance in Definition (1) by

$\displaystyle\operatorname{dist}_{p}((\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n}),(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n}))\coloneqq\left(\sum_{j=1}^{n}\frac{1}{2}(\|\tau_{j}-\omega_{j}\|^{p}+\|f_{j}-g_{j}\|^{p})\right)^{\frac{1}{p}}.$

for some $p>0$.

It is clear that for a (Hilbert space) frame $\\{\tau_{j}\\}_{j=1}^{n}$ for a
Hilbert space $\mathcal{H}$, by defining $f_{j}:\mathcal{H}\ni h\mapsto\langle
h,\tau_{j}\rangle\in\mathbb{K}$, for all $1\leq j\leq n$, Definitions 2.2,
2.3, 2.4, 2.5 and 2.6 reduce to Definitions 1.2, 1.3, 1.4, 1.5 and 1.6,
respectively. In view of Theorems 1.7, 1.8 and 1.9 we ask following three
problems.

######  Problem 2.8.

Whether there is an algorithm for turning every ASF into an equal norm ASF
with same frame operator?

######  Problem 2.9.

What is the closest (in terms of distance given in Definition 2.6) Parseval
ASF to a given ASF?

######  Problem 2.10.

What is the closest equal norm ASF (in terms of distance given in Definition
2.6) to a given equal norm ASF?

Here we state Paulsen problem for Banach spaces.

######  Problem 2.11.

(Paulsen problem for Banach spaces) Find the function
$f:(0,1)\times\mathbb{N}\times\mathbb{N}\to[0,\infty)$ so that for any
$\varepsilon$-nearly equal norm Parseval ASF
$(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ for a d-dimensional Banach
space $\mathcal{X}$, there is an equal norm Parseval ASF
$(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n})$ for $\mathcal{X}$
satisfying

$\displaystyle\operatorname{dist}^{2}((\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n}),(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n}))=\sum_{j=1}^{n}\frac{1}{2}(\|\tau_{j}-\omega_{j}\|^{2}+\|f_{j}-g_{j}\|^{2})\leq
f(\varepsilon,n,d).$

Moreover, whether $f$ depends on $n$?

We can reformulate Problem 2.11 as follows.

######  Problem 2.12.

(Paulsen problem for Banach spaces) Find the function
$f:(0,1)\times\mathbb{N}\times\mathbb{N}\to[0,\infty)$ so that for any
$\varepsilon$-nearly equal norm Parseval ASF
$(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ for a d-dimensional Banach
space $\mathcal{X}$,

$\displaystyle\inf\\{$
$\displaystyle\operatorname{dist}^{2}((\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n}),(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n})):(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n})\text{
is an equal norm Parseval }$ $\displaystyle\text{ASF for }\mathcal{X}\\}\leq
f(\varepsilon,n,d).$

Moreover, whether $f$ depends on $n$?

In view of Theorem 1.12 we first ask the following problem.

######  Problem 2.13.

Does there exists a solution for the Paulsen problem for Banach spaces
(Problem 2.11)?

Here we state projection problem for Banach spaces.

######  Problem 2.14.

(Projection problem for Banach spaces) Let $\mathcal{X}$ be a $d$-dimensional
Banach space with Auerbach basis
$(\\{\zeta_{k}\\}_{k=1}^{d},\\{u_{k}\\}_{k=1}^{d})$
($\\{\zeta_{k}\\}_{k=1}^{d}$ is the collection of dual functionals). Find the
function $g:(0,1)\times\mathbb{N}\times\mathbb{N}\to[0,\infty)$ satisfying the
following: If $P:\mathcal{X}\to\mathcal{X}$ is a projection (idempotent) of
rank $n$ satisfying

$\displaystyle(1-\varepsilon)\frac{n}{d}\leq\|Pu_{k}\|^{2}=\|\zeta_{k}P\|^{2}=|\zeta_{k}(Pu_{k})|\leq(1+\varepsilon)\frac{n}{d},\quad\forall
1\leq k\leq d,$

then there exists a projection (idempotent) $Q:\mathcal{X}\to\mathcal{X}$ with

$\displaystyle\|Qu_{k}\|^{2}=\|\zeta_{k}Q\|^{2}=|\zeta_{k}(Pu_{k})|=\frac{n}{d},\quad\forall
1\leq k\leq d,$

satisfying

$\displaystyle\sum_{k=1}^{d}\frac{1}{2}(\|Pu_{k}-Qu_{k}\|^{2}+\|\zeta_{k}P-\zeta_{k}Q\|^{2})\leq
g(\varepsilon,n,d).$

We already remarked that Paulsen and projection problems were connected using
chordal distance. Even though we are unable to connect Pausen and projection
problems for Banach spaces, we introduce the notion of chordal distance for
subspaces of Banach spaces. Chordal distance between two subspaces of a finite
dimensional Hilbert space is defined using angles between them. Since we do
not have inner product we take alternative way using a characterization to
introduce the notion for Banach spaces.

######  Definition 2.15.

Let $\mathcal{X}$ be a $d$-dimensional Banach space and
$P,Q:\mathcal{X}\to\mathcal{X}$ be rank $m$ projections onto subspaces
$\mathcal{Y}$ and $\mathcal{Z}$ of $\mathcal{X}$, respectively. We define the
chordal distance between $\mathcal{Y}$ and $\mathcal{Z}$ as

$\displaystyle\operatorname{dist}_{\operatorname{chordal}}(\mathcal{Y},\mathcal{Z})\coloneqq(m-\operatorname{Trace}(PQ))^{\frac{1}{2}}.$

######  Problem 2.16.

Whether there is a relation between Paulsen problem for Banach spaces (Problem
2.11) and projection problem for Banach spaces (Problem 2.14)?

We next note that dilation result for a class of ASFs has been derived in
[39]. For finite dimensional spaces it states as follows. We refer [38, 39]
for the notion of approximate Riesz bases for finite dimensional Banach
spaces.

######  Theorem 2.17.

[39] Let $(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ be an ASF for
$\mathcal{X}$. Then there exists a Banach space $\mathcal{Y}$ which contains
$\mathcal{X}$ isometrically and an approximate Riesz basis
$(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n})$ for $\mathcal{Y}$ such
that

$\displaystyle
f_{j}=g_{j}P_{|\mathcal{X}},~{}\tau_{j}=P\omega_{j},\quad\forall 1\leq j\leq
n,$

where $P:\mathcal{Y}\to\mathcal{X}$ is an onto projection. If
$(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ is a Parseval ASF or
Schauder frame for $\mathcal{X}$, then
$(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n})$ can be taken as
$g_{j}(\omega_{k})=\delta_{j,k},\forall 1\leq j,k\leq n$.

Using Theorem 2.17 and based on Theorem 1.16 we can ask following problem.

######  Problem 2.18.

Can Paulsen problem for Banach spaces (Problem 2.11) be solved for $n$ and $d$
which are relatively prime?

In view of Theorem 1.22 we ask the following problem.

######  Problem 2.19.

Can the Paulsen problem for Banach spaces (Problem 2.11) be reduced like
Theorem 1.22 using Theorem 2.17?

We can also formulate the following problems.

######  Problem 2.20.

Classify Banach spaces for which the solution to Paulsen problem for Banach
spaces 2.11 exists.

######  Problem 2.21.

Classify Banach spaces for which solution to projection problem for Banach
spaces 2.14 exists.

######  Problem 2.22.

Classify Banach spaces for which both Paulsen problem 2.11 and projection
problem 2.14 have solutions.

######  Remark 2.23.

A web of conjectures and problems for p-approximate Schauder frames [37, 40]
for Banach spaces have been made by the author in [38]. Note that for finite
collections in finite dimensional Banach spaces, the notions approximate
Schauder frames and p-approximate Schauder frames coincide.

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[2]Mechanical Engineering Department, Faculty of Engineering, University of
Zanjan, Zanjan, Iran 1]Mechanical Engineering Department, Faculty of
Engineering, University of Zanjan, Zanjan, Iran

# A novel floating piezoelectric energy harvesting from water waves: fully-
coupled simulation

Erfan Rajabi Khorramdarreh  Mohammad Mostafa Mohammadi<EMAIL_ADDRESS>Jafar Ghazanfarian  *  [

###### Abstract

A fully-coupled-fluid-structure-piezoelectric model is presented based on the
finite element method that is capable of modeling piezoelectric harvesters in
the presence of free-surface flow and floating lightweight harvesters with
arbitrary movements. The Navier-Stokes equations and the phase-field method
are employed to describe the free-surface waves. Equations of the conservation
of linear momentum in company with the piezoelectric constitutive relations in
the strain-charge form are utilized to obtain solid deformation and the
electric field intensity. According to the results, attaching mass to the tip
of the beam leads to 13.5% rise in the output voltage compared to the state
without the attached mass. Another studied factor was the influence of the
load resistance on voltage and the output power. The generated voltage grows
along with the load resistance until it reaches a constant value. However, the
power has an optimum load resistance that is 2.61 times higher than the
reference state. The beam’s inclination is significant in effectively
exploiting water waves due to raising the root mean square (RMS) value of the
voltage by 89.53% at an angle of 40 degrees relative to the vertical state. By
altering the thickness of the beam from 1 mm to the value of 1.5 mm, the RMS
voltage exhibits a considerable upward change of 66%. By increasing the length
of the cantilever beam connected to the buoyant structure, and therefore, the
indentation of the beam in the water, the output voltage grows, such that a
beam with the length of 25 cm shows a 2.92 times increase in the output
voltage relative to the beam with a length of 15 cm.

###### keywords:

phase-field method, free-surface flow, piezoelectrics, energy harvesting,
floating harvester,

##  1 Introduction

Due to ongoing advancements in microcontrollers and microelectronics,
designing sensors with lower energy consumption is receiving much attention
nowadays, and developments in these devices have outpaced conventional battery
technologies [1, 2]. Using new energy harvesting techniques to acquire
electricity from the environment is a viable answer to the existing problems.
Water flow is ubiquitous in nature and in industries from small to large
scales. Also, they have higher power density potential in comparison with
other sources of renewable energy like solar and wind energy. Water waves give
an opportunity to harness energy from the kinetic energy of flow through
mechanical vibrations due to wave-instruction interaction [3, 4].

Three mechanisms of converting mechanical vibrations to electrical energy have
been proposed: electrostatic [5], electromagnetic transducers [6], and
piezoelectrics [7]. The prime features of piezoelectric materials are their
great power density (generated power with respect to volume), lower cost of
fabrication and simplicity of use along with their versatility in a wide range
of structures and applications [8].

Accompanying Piezoelectric technology with the fluid-solid interaction (FSI)
phenomenon is a viable alternative for small-scale turbines because of its
lower cost and simpler mechanism [9]. When water waves hit an elastic
structure, the imposed mechanical vibration could be converted into
electricity via attached piezoelectric materials to the structure. One of the
most common piezoelectric materials is PZT (lead zirconate titanate)
materials, which contributes to a large energy conversion rate owing to their
high electro-mechanical coupling constants [10].

In recent years, many researchers have studied piezoelectric harvesters buried
in fluid flow. Thus, lots of models and experiments have been designed to
analyze the system’s behavior. The following lines will be dedicated to
reviewing numerical approaches adopted by scholars for studying piezoelectric
energy harvesting from water flows.

Hu et al. [11] proposed an analytical model as well as an experimental
examination of energy harvesting via vortex-shedding vibrations. The vortices
shedding on the vibrating plate and their flow behaviour has been considered
by the Lamb-Oseen vortex model to provide a simpler physical grasp of the
problem. Then, they explored the impact of the Reynolds number and finding
optimum distance for voltage and power generation. Nabavi et al. [12]
introduced an innovative piezoelectric-based ocean waves energy convertor that
can generate power from breaking ocean waves on a vertical face. They also
used classical beam theory accompanied by Lagrange’s method to derive electro-
mechanical formulations analytically and then validated the results by
experimental data.

Sang et al. [13] researched harvesting energy from vortices induced by water
flow on a cantilever harvester with a cylindrical blade attached to its tip.
They proved that expanding the diameter of the cylinder and decreasing its
mass resulted in higher voltage by performing an experiment and providing a
mathematical model using the Euler-Bernoulli equation, which is a distributive
parametric model. An efficient piezoelectric coupled buoy was created by Wu et
al. [14] to harvest energy from ocean waves. Several cantilever beams
containing a piezoelectric material were joined to a buoyant structure,
allowing it to be readily suspended in intermediate and deep ocean waves. They
provided a numerical model using Euler-Bernoulli’s beam theory, which
describes and accounts for forces acting on the buoyant object including the
drag, the lift, and the buoyancy forces.

Guo et al. [15] conducted a two-dimensional numerical simulation to explore
performance of the VIV harvester’s installation depth while considering it as
a mass-spring-damper oscillator and solving governing equations by applying
the Newmark-$\beta$ approach through a UDF code. The simulation findings
revealed that if the installation depth is more than half a meter, the free-
surface of the flow has no effect on the harvester’s performance. Xie et al.
[16] developed a piezo-beam harvester comprised of a cantilever beam with an
attached mass to generate power from the longitudinal waves. They provided a
mathematical model for the resistance and generated voltage by means of the
linear theory of Airy wave and classical elastic beams in order to investigate
the influence of factors like the attached mass to the beam tip, the ratio of
the beam width to the beam thickness, and the wave height. By adding weight to
the tip and raising the mass proportion of weight to the mass of the beam, the
output voltage could be up to 6 times higher than the case without the
attached weight.

In order to examine the generated voltage of various piezoelectric energy
harvester (PEH) arrangements found on vortex-induced oscillations, that is
typically utilized in low-velocity water in most of surroundings, Sui et al.
[17] performed experiments and simulations by considering harvesters as a
single-degree-of-freedom (SDOF) model, while Ramirez [18] demonstrated the
effect of three cylinders operating in tandem arrangement on the voltage.
Yayla et al. [19] used a similar numerical strategy, embedding a piezoelectric
generator behind vortex-generating plates involving nozzles of various angles
and diameters in a single-phase turbulent flow channel. Yamac et al. [20]
simulated a wave tank by employing the moving mesh concept and the volume-of-
fluid technique along with a code containing a similar methodology in Wu et
al.’s article to assess the potency of fluid column height and waves height.

Using magnets and flow-induced vibrations, Zou et al. [21] developed bistable
piezoelectric harvesting process. Piezoharvester magnets are influenced by
driving magnets on cantilever beams. The system is defined as a mass-spring-
damper oscillator. The blade angle and water velocity affect the output power.
As the water velocity rises, so does the driving magnet displacement. Higher
hydrodynamic area and vibration and power are results of an increased blade
angle. Viet et al. [22] developed a mass-spring system with piezoelectricity-
lever devices for vibration amplification to transform wave motion to
mechanical vibration through a floater. A mathematical model stemmed from the
Lagrangian-Euler approach is proposed and resolved using the iterative
technique to determine the root-mean-square value of the output electric power
in order to study factors such as the moving mass in the system and the ocean
wave amplitude. Mirab et al. [23] employed numerical optimization for the
variables that play role in harnessing electricity assisted with taking into
account methods of the linearized wave theory and JONSWAP spectra formula to
enhance the effective electric power generated from a PZT cantilever harvester
restricted at the ocean floor.

Most of the aforementioned numerical works have modeled the energy harvesting
system for single-phase flows, and in order to accomplish this, the
piezoelectric beam is considered as a single-degree-of-freedom system.
However, this method is ineffective at higher working frequencies.
Furthermore, in simulations related to floating harvesters surrounded by the
free-surface flow of water, several constraints are imposed on their movement
so that the methods like finite-element or distributed parameter model can be
implemented. These constraints do not apply to lightweight floaters. As a
consequence, a fully-coupled simulation of the piezoelectric beam with the
two-phase flow (free-surface flow of air-water) could address the mentioned
problems. To the best of author’s knowledge, this is the first attempt to
investigate a fully-coupled simulation of a floating piezoelectric beam in a
free-surface wavy flows. The current study intends to provide a fluid-
structure-piezoelectric model established upon the finite element method in
COMSOL software in order to simulate piezoelectric harvesters interacting with
water waves. The physical conceptualization was established by coupling the
Navier-Stokes equations for incompressible fluid flow, the Gauss law for
circuit and piezoelectricity equations, and the Navier equations for
characterizing the deformation of the elastic beam. The final results is to
develop the design of a harvester floating over water waves as our main
engineering problem, which is so important for remote area and offshore
applications.

The following structure best describes the article. After the literature
review, the configuration feature and the governing equations of a
piezoelectric harvesting system established on water waves are represented in
section 2. Sections 3 and 4 contains the numerical verification and results,
respectively. Finally, section 5 demonstrates the conclusions.

##  2 Governing equations

###  2.1 Problem statement and experimental setup

The harvesting system under study is shown in Fig. 1. The system is comprised
of a stainless steel beam and a piezoelectric layer of PZT-4 mounted on one
side of the beam (unimorph piezoelectric energy harvester). Surface water
waves created by a piston-type wavemaker provide time-varying force on the
cantilever beam, which is needed for the piezoelectric layer to generate
voltage. This layer is paralleled with an electrical resistance and located
near the beam’s support due to being exposed to a higher range of imposed
stresses in comparison to other distances. Fig. 2 and Tab. 1 present the
schematic geometry of the problem and geometrical parameters, respectively.

Figure 1: Configuration of the energy harvesting system. Table 1:  Geometrical properties of the energy harvesting system. Parameter |  Description | Value
---|---|---
H |  Height of the channel | 60 (cm)
L |  Length of the channel | 380 (cm)
${L_{b}}$ |  Length of the beam | 52 (cm)
${t_{b}}$ |  Thickness of the beam | 1 (mm)
${L_{p}}$ |  Length of the piezoelectric | 2.54 (cm)
${t_{p}}$ |  Thickness of the piezoelectric | 0.1 (mm)
R |  Electrical resistance | 1 (M$\Omega$)
${I_{W}}$ |  Wavemaker indentation in water | 4 (cm)
${I_{b}}$ |  Beam indentation in water | 3 (cm)
$\omega$ |  Frequency of wave maker | 1.37 (Hz)
Figure 2: Front view of the 2D geometric model used in the simulations.

The wavemaker, by reciprocating motion, creates a turbulent flow in the
channel. The excitation frequency of the surface waves is small. Also,
concerning the material type of the beam, the use of linear elastic model is
reasonable to simulate the harvesting process. Furthermore, it is supposed
that the beam has no twist and torsional shear stress; thus, it is reasonable
to simulate the problem in two dimensions. Due to the multiphysics essence of
the system, it is vital to reduce the computational cost.

Water and air are considered as Newtonian fluids that are governed by
continuity and momentum equations in unsteady and ALE (arbitrary
Lagrangian–Eulerian) form[24]:

$\frac{{\partial\rho}}{{\partial t}}\;+\;\nabla.\,(\rho u)=0$ (1)
$\rho\frac{{\partial\,u}}{{\partial\,t}}+\rho\,((u-w){\kern 1.0pt}.\nabla
u)=\nabla.{\kern 1.0pt}\,[-pI+K]+F$ (2)

where u and p are the fluid velocity and pressure, w is the mesh velocity,
$\rho$ is the density and F is the external force. The standard
$k-\varepsilon$ model is applied to capture turbulence generated in the
channel, which is commonly used to study wave simulations in such applications
[25].

$K=(\mu+{\mu_{t}})(\nabla u+\nabla{u^{T}})$ (3)
$\rho\frac{{\partial\,\kappa}}{{\partial\,t}}+\rho(u\,.\nabla)\kappa=\nabla.[(\mu+\frac{{{\mu_{t}}}}{{{\sigma_{k}}}})\nabla\kappa]+{P_{k}}-\rho\varepsilon$
(4)
$\rho\frac{{\partial\,\varepsilon}}{{\partial\,t}}+\rho(u\,.\nabla)\varepsilon=\nabla.[(\mu+\frac{{{\mu_{t}}}}{{{\sigma_{\varepsilon}}}})\nabla\kappa]+{C_{\varepsilon
1}}\frac{\varepsilon}{\kappa}{P_{\kappa}}-{C_{\varepsilon 2}}{\kern
1.0pt}\rho\frac{{{\varepsilon^{2}}}}{\kappa}$ (5)
${\mu_{t}}=\rho\,{C_{\mu}}\frac{{{\kappa^{2}}}}{\varepsilon}$ (6)
${P_{\kappa}}={\mu_{t}}[\nabla u:(\nabla u+(\nabla{u^{T}})]$ (7)

The turbulent viscosity $\mu_{t}$ is calculated by k and $\varepsilon$ in Eq.
(6), where $C_{\mu}$ is an empirical coefficient whose value is often deemed
to be 0.09. The model constants are
${C_{1\varepsilon}}=1.44\quad\;{C_{2\varepsilon}}=1.92\quad\;{\sigma_{k}}=1\quad\;{\sigma_{\varepsilon}}=1.3\quad\;$
[26].

The computational grid must be deformed throughout time to capture the changes
in the flow caused by body deflection. In spite of the ever-growing popularity
of the space-time methods in FSI problems, this work resorted to the ALE
methods which are straightforward to implement. The Laplace’s equation governs
the mesh movement.

${\nabla^{2}}w=0\;\;,\;\,on\;{\Omega_{f}}\quad\;\;\;\;w=\frac{{\partial\eta}}{{\partial
t}}\;,\quad on\sum\,(t)$ (8)

In the equation above, $w$ is the grid velocity, and $\eta$ denotes the
displacement of the fluid-solid interface ($\Sigma\,(t)$). To update the
grid’s point for reference and the current mesh deformation, Eq. (9) is used
[27, 28]:

${x^{n+1}}-{x^{n}}=r\Delta t$ (9)

For smoothing the dynamic mesh, the Winslow method yields one coupled system
of equations for all coordinate directions, which is non-linear and inclined
to permit elements to be stretched excessively. Thus, large displacements
before inverting are allowable [29].

###  2.2 The phase-field method

In order to track the air/water interface (two immiscible fluids), a diffuse-
interface technique named the phase-field method is applied to capture the
exact position of the interface by mixing the free energy. This method can
handle complicated morphological changes in the interface, and does not
necessitate calculating interfacial curvature, which is usually challenging.
An auxiliary variable $\phi(x,t)$ is employed to trace the air-water
interface. $\phi$ = 1 and $\phi$ = -1 present the pure air and pure water,
respectively. This function changes smoothly around the interface between 0
and 1. The dependence of viscosity and density on the phase-field function
leads the Navier-Stokes equations coupled with the convective Cahn-Hilliard
equation describing the evolution of the phase-field function [30, 31, 32].
The Cahn-Hilliard equation in the ALE framework is as follows:

$\frac{{\partial\,\phi}}{{\partial\,t}}+(u-w)\,.\nabla\phi=\nabla.\,\gamma\,\nabla
G$ (10) $G=\lambda\,[-{\nabla^{2}}\phi+\frac{{\phi\,({\phi^{2}}-1)}}{s}]$ (11)

where G and $\phi$ are chemical potential and the dimensionless phase-field
function, respectively. The phase-field interface decomposes the fourth-order
PDE in Eq. (10) into two second-order PDEs:

$\frac{{\partial\,\varphi}}{{\partial\,t}}+(u-w).\nabla\phi=\nabla.\frac{{\lambda\gamma}}{{{s^{2}}}}\nabla\psi$
(12) $\psi=-\nabla.\,{s^{2}}\nabla\phi+{\phi}\,({\phi^{2}}-1)$ (13)

where $\gamma$, $\lambda$, $s$ symbolize the mobility of the interface, the
mixing energy density, and a controlling interface parameter, respectively.
Half of the largest mesh size passing through the interface was prescribed as
a controlling parameter. The mobilizing parameter sets the time scaling for
the Cahn-Hilliard equation. It should be both enough large or small to keep a
fixed thickness in the interface while preventing overly damping of the
convective terms. $\gamma$ has an empirical value for the mobility of the
interface in the Cahn-Hilliard equation. Dong and Shen [33] presented details
about the influence of $\gamma$ in their tests, finding that when its value is
less than $10{\,{}^{-4}}$, it has no impact on the outcomes of the capillary
wave study. In this work, given the maximum surface velocities created by the
wavemaker, the value of $5\times{10^{\,-5}}$ is considered [34]. The following
Eqs. (14) connect the dynamic viscosity, $\mu$, and the density, $\rho$, to
the phase-field variable.

$\rho(\phi)=\frac{{{\rho_{1}}+{\rho_{2}}}}{2}+\frac{{{\rho_{1}}-{\rho_{2}}}}{2}\phi\;\quad,\quad\mu(\phi)=\frac{{{\mu_{1}}+{\mu_{2}}}}{2}+\frac{{{\mu_{1}}-{\mu_{2}}}}{2}\phi$
(14)

Imposing initial and boundary conditions on the fluid domain is of great
significance. One of these boundary conditions is the continuity of velocities
on the solid interface; the other represents a wetted wall. When the waves
intersect the walls in the computational domain, it is required to specify the
contacting angle, ${\theta_{w}}$. For simplicity, a contact angle of
${90^{o}}$ was appointed [35]. Also, the mass flow across the walls equals
zero:

$n.\,\frac{{\lambda\gamma}}{{{s^{2}}}}\nabla\psi=0$ (15)

To specify the initial position of the boundary between two fluids, the phase-
field variable could be specified explicitly for a time-dependent study to be
run directly. The initial conditions are given by:

$u(x,0)={u_{0}}\,(x)\quad,\quad\phi(x,0)={\phi_{0}}\,(x)$ (16)

###  2.3 Structure

The linear momentum conservation equations govern the structure deformation.
The solid problem describes the beam dynamics through an isotropic linear
elastic model:

${\rho_{s}}\frac{{{\partial^{2}}u_{s}}}{{\partial\,{t^{2}}}}=\nabla.\,{\sigma_{s}}+{\rho_{s}}{b_{s}}$
(17)

where $u_{s}$ is the solid displacement vector, $\rho_{s}$ denotes the solid
density and $\sigma_{s}$ is the Cauchy stress tensor, which is defined as
follows:

${\sigma_{s}}=c:\varepsilon_{s}$ (18) $\varepsilon_{s}=\frac{1}{2}[\nabla
u_{s}+(\nabla u_{s}){{\kern 1.0pt}^{T}}]$ (19)

in which c is the linearized Young’s modulus, and $\varepsilon_{s}$ is the
strain tensor obtained according to Eq. (19).

The fluid motion equations, the structure deformations laws, and the
transmission conditions at the fluid-structure interface characterize a fluid-
solid interaction. The solid-fluid interface was assigned kinematic and
dynamic constraints to form the FSI coupling [36].

$\frac{{\partial\,\eta}}{{\partial\,t}}=u$ (20) ${\sigma_{f}}.\,{\kern
1.0pt}n+{\sigma_{s}}.{\kern 1.0pt}\,n=0$ (21)

Eq. (20) shows the equity of solid and water velocity at the interface (also
implies the no-slip boundary condition). In addition, Eq. (21) describes the
force balance at the interface. $\eta$ and $n$ are the interface displacement
and the outward normal vector of the solid boundary, respectively.

###  2.4 The piezoelectric effect

Piezoelectricity is a property of specific crystalline materials that could
convert an applied load to electricity. In other words, it is a coupled
behaviour of strain change and electrical charge due to an applied electric
and stress fields. The piezoelectric poled through the thickness could be
modeled by two linearized constitutive equations in the strain-charge form:

$\varepsilon_{s}={s_{e}}\,{\sigma_{s}}+{d^{T}}{E_{p}}$ (22)

$D=d\,{\sigma_{s}}+{\varepsilon_{T}}\,{E_{p}}$ (23)

where $D$ is the electric displacement field, $E$ is the electric field, and
$\varepsilon_{T}$ is the dielectric permittivity. $d$, $d^{T}$, $s_{e}$ are
the elastic compliance matrix, the piezoelectric coefficient, and the
piezoelectric constants, respectively. In order to complete the electro-
mechanical coupling, it is necessary to relate the piezoelectric equations
with the circuit through an electrostatic module. An electrostatic module is
not a time-dependent physical aspect unless coupled with other modules.
Through Gauss’s law, this coupling is possible. The piezoelectric thickness is
subjected to the zero charge boundary condition, while the piezoelectric-beam
interface has the ground boundary condition. Furthermore, the other side of
the material is bounded by the terminal boundary condition to obtain the
generated voltage by the PZT material through a paralleled load resistance
[37, 38].

###  2.5 Grid/time-step size independence tests

The grid independence procedure has been carried out by use of non-uniform
unstructured meshes refined near the beam and the air-water interface (Fig.
3). In the wave-structure interaction process modeling, the whole boundary
layer and the fluid-fluid interface is refined, and the sensitivity of the
computation results for the pressure distribution on the beam (Fig. 4), the
maximum deflection of the beam, and the maximum output voltage have been
examined by exercising the grids with total elements of 99631, 173986, 314520
in Tab. 2. These three computational grids have been generated by
concentrating on refinement near the beam and the air-water interface. The
results of the grid with 173986 and 314520 elements including the pressure
distribution shows a favorable agreement. Also, the maximum deflection of the
beam along with the maximum output voltage, the computational outcomes have
been converged with less than 1% difference. Consequently, the domain with
173986 elements was adopted as the optimum grid.

Table 2:  Examination of the grid independence test for the maximum output voltage Grid | Coarse | Fine | Finest
---|---|---|---
Maximum output voltage (V) | 0.343 | 0.381 | 0.385
Figure 3: The computational grid and a close snapshot of the beam. Figure 4:
The pressure distribution on the front side of the beam obtained using three
computational grids.

It is feasible to use an adaptive time-step computation determined based on
the relative and absolute tolerances. By considering the solver convergence,
if larger time-steps are selected, they do not affect the final solutions
because the accuracy is determined according to these tolerances. The relative
and absolute tolerances are set to 0.01 and 0.001, respectively. The implicit
BDF (backward differentiation formula) with manual time-stepping is applied in
the fully-coupled solver, and automatic Newton settings in the solver helps
prevent divergence. As seen in Tab. 3, two time-steps have nearly generated
the same results for two studied parameters. Thus, in order to decrease the
costs of computations, the time-step size of 0.001 has been employed.

Table 3: Investigation of the time-step size independence for two parameters of maximum output voltage and maximum deflection of the beam. Time-step size |  Maximum output voltage |  Maximum deflection of the beam
---|---|---
0.0005 |  0.379 |  5.59
0.001 |  0.381 |  5.65
0.005 |  0.394 |  5.02

##  3 Validation/verification

To verify the accuracy of the numerical model presented in the previous
section, results have been compared with Valhorne’s numerical outcomes [39]. A
2D benchmark dam break involving an elastic beam was considered in Valhorne’s
numerical study to provide the deflection of the beam tip over time as well as
the volume fraction contours. In Valhorne’s study, a water column strikes an
elastic beam while flowing under effect of its own weight and leads to the
beam’s deflection. The interaction between the beam and fluid flow also
affects the free-surface shapes. Contours of he volume fraction of water
extracted from the simulation is compared with the free-surface flow figures
provided in Valhorne’s numerical study. A good agreement is seen in Fig. 5 for
the deflection of the beam and the water tongue obtained from the presented
simulations and the mentioned numerical study.

Figure 5:  Fluid volume contours and deflection of the beam, (a) the presented
study, (b) the numerical study of [39] for the dam break problem.

Also, a laboratory set-up of the harvesting system was constructed to validate
the output RMS voltage with those of the presented model. Configuration of the
energy harvesting in the experiments was presented in Fig. 1. The
piezoelectric beam with indentation of 3(cm) in water produces 76mV output RMS
voltage while from the numerical simulations this value is 83 mV, which shows
a 9.18% difference as seen in Fig. 6.

Figure 6: The time-history of the voltage and comparison with the
experimentally obtained RMS voltage.

##  4 Results and discussion

###  4.1 Added mass to the tip of the beam

The key to harvesting piezoelectric energy is the low operation frequency of
this harvester owing to the frequency of surface waves at 1.37 Hz. Hence, the
beam’s first vibration mode must be stimulated to harness energy. Adding mass
to the beam decreases the damped natural frequency, thus it is more likely to
happen the resonance phenomenon. To examine the effect of adding mass to the
beam tip on the output voltage, a piezoelectric beam with a thickness of 1 mm
and a distance of 40 cm from the wavemaker was positioned and four masses
weighing 50, 100, 130, and 180 grams were added to the beam tip. Given the
constitutive relations provided in Section 2.4, the fact that the electric
displacement is commensurate with the stress means that the rise in the stress
range leads to generation of a higher RMS voltage. Enhancement of the
displacement range of the beam tip could indicate an increase in the stress
range and the generated voltage due to the state of the beam’s constraint
(cantilever).

This phenomenon is important from the point of view that the tip displacement
increases for the same bending force. The output voltage increases with adding
the mass up to 100 grams compared to the state without the mass, and this rise
is up to 13.5% for the mass of 100 grams according to the data in Fig. 7. The
rise in voltage is brought about by a decrease in the structure’s damped
natural frequency and its approach to the frequency of the surface waves. The
voltage begins to drop for the weights between 120 grams and 180 grams as a
result of moving away from the resonant frequency. The RMS voltage for the
mass of 180 grams attached to the beam tip is 17% less than the beam without
the mass. As shown in Fig. 8, the range of displacement for the mass of 100
grams is higher than the other states. This trend confirms the above-mentioned
statements.

Figure 7:  The RMS output voltage from the piezoelectric beam with and without
the mass attached to the tip. Figure 8:  Temporal variation of the
displacement of the beam tip with and without the attached mass.

###  4.2 The effect of the load resistance

In all simulations, the electrical resistance of the series load with the
piezoelectric converter circuit is equal to 1 $M\Omega$. By altering the load
resistance value in accordance with Patel et al. [40] work, the RMS voltage
and the electric power both change. As seen in Fig. 9, for higher values of
the load resistance, the effective voltage raises in a non-linear manner so
that the slope of the graph decreases and the RMS value eventually tends to a
constant value, which is justifiable due to the data obtained from Eq. (24)
[40].

${U_{RL}}(t)=\frac{{(1/{R_{L}})}}{{1/\sqrt{{{({R_{L}}+{R_{S}})}^{2}}+{{(\frac{1}{{\omega
C}})}^{2}}}}}\times{U_{p}}(t)$ (24)

Figure 9:  (a) The output power and (b) the RMS voltage for different load
resistances.

where $U_{RL}$, ${R_{L}}$, ${R_{S}}$, and ${U_{p}}(t)$ are the generated
voltage through a specified load resistance, the load resistance in series
with the circuit, the internal resistance of the piezoelectric material, and
the generated voltage of the harvester, respectively. However, the electric
power increases up to an optimum value for higher values of the load
resistance, and then begins to drop off once the optimum resistance is
reached. Eq. (25) shows that the resistance and the voltage have an inverse
relationship thus, there should be an optimum resistance for the generated
voltage [40]. As seen in Fig. 9, the output power for 5 $M\Omega$ load
resistance is 0.0356, which indicates a 2.61 times increase with respect to
the reference resistance that is 1$M\Omega$ .

$<P>{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}=\frac{{{U^{2}}_{RL}(t)}}{{{R_{L}}}}$ (25)

###  4.3 Inclination angle of the beam in water

This section examines influence of the inclination angle of the beam in water
on voltage generation. If the weight of the beam is substantial, the
structure’s orientation will play a critical role in the damped natural
frequency. In the present study, the weight of the structure is not
considerable, but it is possible to boost the efficacy of the transducer from
the wave force by modifying the inclination (the tilt angle) of the cantilever
beam in water relative to the vertical axis. In order to assess the effect of
the angles on voltage, the distance from the wavemaker and the indentation of
the beam in water are kept constant in comparison to the reference state.

It is found that the output RMS voltage for four different inclination angles
of the piezoelectric beam in water has risen relative to the vertical state
(${0^{\,o}}$) so that for angles of ${20^{\,o}}$, ${40^{\,o}}$ and
${60^{\,o}}$, there are increased generated voltage of 43.2%, 89.53% and
68.7%, respectively. The explanation for these increases in the RMS voltage is
demonstrated in Fig. 10, which provides the displacement of the beam tip for
corresponding angles. At the angle of ${40^{\,o}}$ , the largest displacement
range of any other angle is observed; furthermore , the ${60^{\,o}}$ angle
possesses a displacement range that is greater than the others and smaller
than ${40^{\,o}}$. Consequently, the RMS voltage corresponding to ${60^{\,o}}$
is higher than that of two other angles but less than the voltage for
${40^{\,o}}$.

Figure 10:  Displacement of the beam tip for four different inclination
angles.

The size of structure plays a critical role in analyzing the exerted force of
water waves on the piezoelectric-beam transducer. If the ratio of the
structure’s characteristic size (for example, its diameter) to the wave
amplitude is less than 0.2, the structure is deemed small; thus, the drag and
the inertial forces acting on it become significant. The forces acting on
small marine structures, according to Morrison’s equation, are the sum of the
drag and the wave inertial forces [41], which are as follows:

$F=\,\,{F_{Drag}}+{F_{Inertia}}=0.5\,\rho\,\,{C_{D}}\,V\,\,|V|{\kern
1.0pt}{\kern 1.0pt}\,{A_{p}}+{C_{M}}\,\rho\,\,{V_{vol}}\,\dot{V}$ (26)
${C_{M}}={C_{a}}+1$ (27)

where ${C_{D}}$, ${C_{M}}$, ${A_{p}}$, and ${V_{vol}}$ are the drag
coefficient, the inertia coefficient, the projected surface area, and the
volume of the structure, respectively. Both the drag and the inertial forces
applied to the piezoelectric-beam transducer will change as the direction of
the beam differs, but the drag force, which results from the change in the
angle of inclination in the water, is not very effective in raising the
voltage of the harvester. Instead, the inertial force plays a key role in
increasing the forces on the beam and the range of stresses (or displacement).
${C_{a}}$ is the added mass coefficient, which is related to the inertia
coefficient according to Eq. (27). This coefficient is a measure of the
accelerated surrounding fluid around the beam. Fig. 10 implies that by
inclining the beam, the volume of the accelerated surrounding fluid around the
beam is increased relative to the reference state (${0^{\,o}}$), which leads
to the growth of vibrations and the generated RMS voltage in the transducer.
As a consequence, the inertia coefficient is sensitive to the orientation of
the beam.

###  4.4 Thickness of the beam

The impact of four different beam thicknesses on voltage generation has been
studied. By reducing the thickness of the beam from 1 mm (the reference state)
to 0.7 mm, the output voltage decreases by 18.5% while thickening by half a
millimeter leads to 59.2% rise in the output voltage. To find the reason
behind this trend, it is viable to examine the stress that appears in the
piezoelectric material at a point near the end of the beam. Given the obvious
differences of the stress range at the end of the beam, this point is chosen
to analyze the impact of thickness. Fig. 11 shows that the range of stress for
the beam with the thickness of 1.5 mm is greater than the other two cases,
providing evidence of the origin of the rise in the output voltage. Although
the state with the greatest output voltage is associated with the beam with a
thickness of 2 mm, it demonstrates only a 5.81% increase in the output voltage
compared to the beam with a thickness of 1.5 mm. When the static and dynamic
loadings are applied, the stress at the interface of the beam and the
piezoelectric material has a direct relationship with the distance from the
neutral axis of the beam, and an inverse relationship with the second moment
of area of the beam cross-section. According to Fig. 11, the difference in the
range of stress between these two states is not noticeable. This shows that
the increase in the second moment of area of the beam cross-section has
prevented the effect of increasing the distance from the neutral axis to raise
the stress range and the output voltage.

Figure 11:  Temporal variations of the imposed stress in the beam
corresponding to four different thicknesses.

###  4.5 Floating harvester

This section investigates the effect of connecting the piezoelectric beam to a
floating structure with three different lengths, a schematic view as well as
the free-surface interaction with the floater, and the time-history of the
output voltage are given in Fig. 12 and Fig 13, respectively. According to the
dimensions and the weight of the floating structure, three beams with sizes
15, 20, and 25 cm have been connected to the floating structure in the form of
a cantilever beam. The RMS voltage for a 15 cm-long beam (5 cm indentation) is
55.8 mV, and as the length of the transducer beam and its indentation in the
water get larger, so does the output voltage. Such that for a beam with the
length of 25 cm (15 cm indentation), the output voltage is 161.94 mV, equal to
an increase of 2.92 times as seen in Fig. 14.

Figure 12:  The schematic view of the floating harvester (left), the free-
surface interaction with the floater (right). Figure 13:  The time-history of
the generated voltage from the floating harvester. Figure 14:  The RMS of the
output voltage for three indentations of the floating beam in water.

Fig. 15 reveals the origin of the trends in output power generation. It
demonstrates that the lowest stress is related to the small indentations and a
15 cm indentation has a higher stress range in comparison to the other two
lengths, thereby enhancement of strength of the electric field and electric
displacement. It is initially assumed that the beam with smaller indentations
would be stimulated more than the longer ones and a higher voltage may be
generated. However, the longer the length, the less stiff the beam and the
lower its first mode’s natural frequency. The beam’s natural frequency is
inversely related to the second power of the length of the beam.

Eq. (28) is related to the resonant frequency of a cantilever beam in which L,
w, and E are the beam’s length, width and the Young modulus, respectively.
Also, I is the second moment of area for the beam cross-section. ${k_{n}}$ is
equal to 3.52 for the first frequency mode.

$f=\,\frac{{{k_{n}}}}{{2\pi}}\sqrt{\frac{{EI}}{{w{L^{4}}}}}$ (28)

In addition, in this particular case, a longer beam means a deeper penetration
of the beam inside the water, which causes the added mass to surround the beam
and reduces the frequency of the first mode of the beam even more. As the
first frequency mode of the beam gets closer to the excitation frequency of
the water waves (which is always lower than the natural frequency of the
cantilever beam) within the desired bandwidth, it is clear that the beam will
get a higher output voltage with higher depth and length.

Figure 15:  The imposed stress range for three indentations of the floating
beam in water.

##  5 Conclusion

In this study, a fully-coupled fluid-structure-piezoelectric model was
suggested to investigate the energy harvesting from water waves. The electro-
mechanical formulations and two-phase flow modeling were coupled in the
framework of the finite-element approach. The prime features of the fully-
coupled modeling are the ability of simulating piezoelectric harvesters at the
presence of the free-surface flow (two-phase flow) and the lightweight
floaters with arbitrary movements. The results are verified with the numerical
study and a fabricated sample in the lab.

Therefore, following achievements can be highlighted:

  * •

To approach the damped natural frequency of the structure and the excitation
frequency of the water waves, a mass of 100 grams was added to the tip of the
beam, and as a result, the voltage increased by 13.5%.

  * •

The influence of electrical resistance on the output voltage and the power was
studied. The output voltage increased when the load resistance grew until it
reached a constant value. However, itbis found that the power had an optimal
load resistance, for which the power enhanced by 2.61 times relative to the
reference state.

  * •

The impact of the inertial force of water waves on the beam is accentuated by
raising the inclination angle of the beam within the water relative to the
vertical axis. At an angle of 40 degrees, the maximum output voltage is
attained by an increase of 89.53%.

  * •

The output voltage for the 1.5 mm thickness grew by 59.2 percent compared to
the 1 mm thickness, while the output voltage for the 2 mm thickness increased
by just 5.81 percent compared to the 1.5 mm thickness.

  * •

Longer indentations of the beam connected to the floater leads to the rise of
the generated voltage such that the beam with the length of 25 cm (15 cm
indentation) shows a 2.92 times increase in the output voltage compared to the
15 cm long beam (5 cm indentation).

##  6 Acknowledgment

This research was supported by the Iran National Science Foundation (Grant
num- ber 98017606).

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