Bending of plates with transverse shear deformation: The Robin problem

1 INTRODUCTION

Many times, solutions of boundary value problems for a mathematical model cannot be computed explicitly, but they can be approximated within acceptable tolerances by means of expansions in a complete set of functions in a Banach space such as L². The method acquires additional interest and usefulness when the choice of these functions is based on the structure of the layer potentials associated with the problem. This type of expansion may encounter problems when the set in question is orthonormalized by, for example, the Gram-Schmidt technique. In this paper, we propose an algorithm that bypasses orthonormalization issues and yields highly promising numerical results within a prescribed accuracy. We illustrate this technique by solving an interior Robin problem for the system governing the bending of elastic plates with transverse shear deformation. Problems with the Dirichlet and Neumann boundary conditions have been solved in our previous works.1, 2

The Robin problem for the model of bending consists of the partial differential equation system and boundary conditions

(1)

where

T(∂₁,∂₂) is the boundary moment-stress operator3, is a vector characterizing the displacements, is the unit vector of the outward normal to ∂S, σ ∈ C^0,α(∂S), α ∈ (0,1), is a symmetric, positive definite 3 × 3 matrix function on ∂S, and is a prescribed 3 × 1 vector function.

It is shown in the work of Constanda3 that problem 1 has a unique solution for any , α ∈ (0,1). The proof of this assertion involves the matrix of fundamental solutions D(x,y) computed in the work of Constanda3 and the associated matrix

The Somigliana representation formula3 for this problem leads to the equalities

(2)

where

(3)

2 COMPUTATIONAL ALGORITHM

Let ∂S_∗ be a simple, closed, C²-curve surrounding S∪∂S, and let

be a set of points densely distributed on ∂S_∗. We construct the vector functions

(4)

where D^(j) are the columns of D. It is easily seen that θ^jk(x) are the rows of P(x^(k),x).

The set

where the 3 × 1 vectors f⁽ⁱ⁾ form a basis for the space of rigid displacements,3 is linearly independent on ∂S and complete in L²(∂S). We reorder the vector sequence

as

and orthonormalize it to obtain a new sequence .

Seeking an approximation of the unknown vector function ψ in the form

(5)

where the are the orthonormalization coefficients and ⟨· ,·⟩ is the inner product on L²(∂S), we use 2 to construct the approximate solution

(6)

We replace by x^(k) in 3 to arrive at

(7)

This equality in combination with 4 and 3 yields

or, what is the same,

(8)

In addition, for i = 1,2,3,

(9)

This computational scheme shows that the approximation u⁽ⁿ⁾ is given by 6, with ψ⁽ⁿ⁾ determined from 5, ⟨θ⁽ⁱ⁾,ψ⟩ from 8 and 9, and L_j(x^(k)) from 7. Detailed analysis indicates that the sequence u⁽ⁿ⁾ converges uniformly in the L²-norm on any closed subdomain of S to the solution u of the problem.

3 NUMERICAL ILLUSTRATION

We consider a numerical example where S is the disk of radius 1 centered at the origin, ∂S_∗ is the circle of radius 2 that is also centered at the origin,

and, for each n = 1,2,…,

As boundary data, we take (in polar coordinates)

This vector function generates the exact solution (in S)

The set is orthonormalized by three different procedures: the classical Gram-Schmidt (CGS), the modified Gram-Schmidt (MGS), and the Householder reflections (HR); see the work of Trefethen.4 We also use a fourth technique, which, based on row reduction, makes orthonormalization unnecessary. Below are the graphical results of the computation.

Figures 1 to 3 display the three components of u⁽⁶³⁾ in S. Figures 4 to 6 show the corresponding computational errors for the approximate boundary values of u⁽⁶³⁾. The curves in Figure 7 are the graphs (in terms of the polar angle) of the computed traces of the components of u⁽⁶³⁾ on ∂S. The graph in Figure 8 indicates the computed error in the traces on ∂S of the components of u⁽⁶³⁾.