Minimizing Eigenvalues for Inhomogeneous Rods and Plates

Abstract

Optimizing eigenvalues of biharmonic equations appears in the frequency control based on density distribution of composite rods and thin plates with clamped or simply supported boundary conditions. In this paper, we use a rearrangement algorithm to find the optimal density distribution which minimizes a specific eigenvalue. We answer the open question regarding optimal density configurations to minimize k-th eigenvalue for clamped rods and analytically show that the optimal configurations are distinct for clamped rods and simply supported rods. Many numerical simulations in both one and two dimensions demonstrate the robustness and efficiency of the proposed approach.

Appendices

Appendix 1

In one dimension, for simplicity, we choose $D=[-1,1]$, and define a uniform grid of points $x_{i}=-1+ih$ where h is the mesh size, $0\le i\le N$ and $N=2/h.$ The discretized eigenfunction is denoted by U in the form of a column vector $\left( U_{0},\ldots ,U_{N}\right) ^{T}$. We approximate the fourth order derivative at $x_{i}$ by the central difference formula

$$\begin{aligned} U_{i}^{''''}\approx \frac{U_{i-2}-4U_{i-1}+6U_{i}-4U_{i+1} +U_{i+2}}{h^{4}}, \end{aligned}$$

for $i=2,\cdots ,N-2$. To approximate the derivatives at $x_{1}$ and $x_{N-1}$, values at ghost points $x_{-1}=-1-h$ and $x_{N+1}=1+h$ are necessary and can be obtained by the given boundary conditions. If clamped boundary conditions are imposed, that is,

$$\begin{aligned} u(-1)=u(1)=u^{'}(-1)=u^{'}(1)=0, \end{aligned}$$

we choose $U_{0}=U_{N}=0$ at two end points and $U_{-1}=U_{1}$ and $U_{N+1}=U_{N-1}$ at two ghost points. Thus

$$\begin{aligned} U_{1}^{''''}\approx \frac{7U_{1}-4U_{2}+U_{3}}{h^{4}}, \end{aligned}$$

and

$$\begin{aligned} U_{N-1}^{''''}\approx \frac{U_{N-3}-4U_{N-2}+7U_{N-1}}{h^{4}}. \end{aligned}$$

The hinged boundary conditions

$$\begin{aligned} u(-1)=u(1)=u^{''}(-1)=u^{''}(1)=0 \end{aligned}$$

lead to $U_{0}=U_{N}=0$ at the boundaries, $U_{-1}=2U_{0}-U_{1}$ and $U_{N+1}=2U_{N}-U_{N-1}$ at two ghost points. Thus

$$\begin{aligned} U_{1}^{''''}\approx \frac{5U_{1}-4U_{2}+U_{3}}{h^{4}}, \end{aligned}$$

and

$$\begin{aligned} U_{N-1}^{''''}\approx \frac{U_{N-3}-4U_{N-2}+5U_{N-1}}{h^{4}}. \end{aligned}$$

Consequently, the matrix representing the biharmonic operator on $[-1,1]$ is formed by assigning the coefficients in the approximation formula of $U_{i}^{''''}$ to the i-th row.

Appendix 2

In two-dimensional rectangle $D=[-a,a]\times [-b,b]$, define a uniform grid of points $x_{i,j}=(x_{1i},x_{2j})$ where $x_{1i}=-a+ih_{1}, x_{2j}=-b+jh_{2}$ where $h_{1}$ and $h_{2}$ are mesh sizes in $x_{1}$- and $x_{2}$- directions, respectively. For simplicity, we assume that $h_{1}=h_{2}=h$. Let $\left( U_{i,j}\right) _{0\le i\le N,0\le j\le M}$ be the matrix of the discretized eigenfunction. The second order central difference scheme involving 13-point stencils is used to approximate the biharmonic operator

$$\begin{aligned} \Delta ^{2}U_{i,j}\approx \frac{1}{h^{4}}\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} &{} &{} +U_{i,j+2}\\ &{} +2U_{i-1,j+1} &{} -8U_{i,j+1} &{} +2U_{i+1,j+1}\\ +U_{i-2,j} &{} -8U_{i-1,j} &{} +20U_{i,j} &{} -8U_{i+1,j} &{} U_{i+2,j}\\ &{} +2U_{i-1,j-1} &{} -8U_{i,j-1} &{} +2U_{i+1,j-1}\\ &{} &{} +U_{i,j-2} \end{array}\right] \end{aligned}$$

for $2\le i\le N-2$, and $2\le j\le M-2$. With clamped boundary conditions the biharmonic operator along $i=1$ is approximated by

$$\begin{aligned} \Delta ^{2}U_{1,j}\approx \frac{1}{h^{4}}\left[ \begin{array}{l@{\quad }l@{\quad }l} +U_{1,j+2}\\ -8U_{1,j+1} &{} +2U_{2,j+1}\\ +21U_{1,j} &{} -8U_{2,j} &{} U_{3,j}\\ -8U_{1,j-1} &{} +2U_{2,j-1}\\ +U_{1,j-2} \end{array}\right] ,\quad 2\le j\le M-2, \end{aligned}$$

and

$$\begin{aligned} \Delta ^{2}U_{1,1}\approx \frac{1}{h^{4}}\left[ \begin{array}{l@{\quad }l@{\quad }l} +U_{1,3}\\ -8U_{1,2} &{} +2U_{2,2}\\ +22U_{1,1} &{} -8U_{2,1} &{} U_{3,1} \end{array}\right] . \end{aligned}$$

The approximating formulas along $i=N-1, j=1$ or $j=M-1$ can be derived similarly. All points at the boundaries are taken as zero, $U_{0,j}=U_{N,j}=U_{i,0}=U_{i,M}=0.$ For the hinged boundary conditions, the discretization is almost the same, except the approximations for points near the boundaries ($i=1$ or $N-1, j=1$ or $M-1$). For example,

$$\begin{aligned} \Delta ^{2}U_{1,j}\approx \frac{1}{h^{4}}\left[ \begin{array}{l@{\quad }l@{\quad }l} +U_{1,j+2}\\ -8U_{1,j+1} &{} +2U_{2,j+1}\\ +19U_{1,j} &{} -8U_{2,j} &{} U_{3,j}\\ -8U_{1,j-1} &{} +2U_{2,j-1}\\ +U_{1,j-2} \end{array}\right] ,\quad 2\le j\le M-2, \end{aligned}$$

and

$$\begin{aligned} \Delta ^{2}U_{1,1}\approx \frac{1}{h^{4}}\left[ \begin{array}{l@{\quad }l@{\quad }l} +U_{1,3}\\ -8U_{1,2} &{} +2U_{2,2}\\ +18U_{1,1} &{} -8U_{2,1} &{} U_{3,1} \end{array}\right] . \end{aligned}$$

The discretization along the other sides can be obtained similarly. Each stencil approximating $\Delta ^{2}U_{i,j}$ is assigned into a row to form the matrix of the discrete biharmonic operator. Therefore, the size of the matrix to approximate the biharmonic operator on a rectangle is $(N-1)(M-1)\times (N-1)(M-1)$.

Appendix 3

For the biharmonic eigenvalue problem on a circular or annular domain, we perform the numerical discretization after transforming the problem into the polar coordinates, and the harmonic operator in terms of $(r,\theta )$ is

$$\begin{aligned} \Delta u =\frac{\partial ^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\frac{\partial ^{2}u}{\partial \theta ^{2}}. \end{aligned}$$

Assume the domain $\Omega$ is a disc with radius one, denoted by D(0, 1). In polar coordinates $(r,\theta )$, the mesh is set to be $r_{i}=\frac{2i-1}{2N+1}, \theta _{j}=\frac{2\pi j}{M}, i=1,2,\ldots ,N+1, j=1,2,\ldots ,M$ to avoid $\left( 0,0\right)$. Suppose $U_{(N+1)\times M}$ is the matrix of discretized eigenfunction, then

$$\begin{aligned} \Delta U_{ij}= & {} \left( \frac{(2N+1)^{2}}{4(2i-1)} +\frac{(2N+1)^{2}}{4}\right) U_{i+1,j}+\left( -\frac{(2N+1)^{2}}{4(2i-1)} +\frac{(2N+1)^{2}}{4}\right) U_{i-1,j}\\&+ \, \left( \frac{M(2N+1)}{2\pi (2i-1)}\right) ^{2} \left( U_{i,j+1}+U_{i,j-1}\right) -\left( \frac{(2N+1)^{2}}{2} +2\left( \frac{M(2N+1)}{2\pi (2i-1)}\right) ^{2}\right) U_{ij}. \end{aligned}$$

Near the center, the ghost point $U_{0,j}$ satisfies

$$\begin{aligned} U_{0,j}=U_{1,j+\frac{M}{2}}. \end{aligned}$$

For clamped boundary conditions, we can define ghost points outside $r=1$ as

$$\begin{aligned} U_{N+2,j}=U_{N,j}. \end{aligned}$$

The simply supported boundary conditions in terms of polar coordinates are written as

$$\begin{aligned} u = \frac{\partial ^{2}u}{\partial r^{2}}+\nu \left( \frac{1}{r^{2}}\frac{\partial ^{2}u}{\partial \theta ^{2}} +\frac{1}{r}\frac{\partial u}{\partial r}\right) =0, \end{aligned}$$

where $\nu$ is a given constant. Thus we can define

$$\begin{aligned} U_{N+2,j} = \frac{(\nu h-2)U_{N,j}+4U_{N+1,j}}{\nu h+2} \end{aligned}$$

as the ghost points outside $r=1$. Let L be the discrete operator $\Delta$ with clamped or simply supported boundary conditions, then the biharmonic operator is approximated by $L^{2}$ after eliminating the last M rows and columns corresponding to the boundary.

The discretization on an annular domain with an inner radius $r_{in}$ and an outer radius $r_{out}$ is similar to the circular case, except that the origin is not included in the domain and therefore the discretized mesh starts at $r_{in}$ and ends at $r_{out}$ in the r-direction . An annulus has both inner and outer boundaries, and therefore $L^{2}$ is obtained by deleting the first and last N rows and columns from the discrete version of the biharmonic operator with clamped or simply supported boundary conditions.