3.4 The Problem of Ill- Conditioning in Solid and Structural Mechanics 28
3.4.3 Physical Identification of Ill-Conditioned Systems
It is apparent from the above discussion that the condition of a matrix is a well defined mathematical quantity that plays an important role in the solution of Kx = f. However, the computationof the spectral condition number from equa
tion (3.37) is, in general, prohibitively expensive. It is therefore important that the engineering numerical analyst should have an understanding of the physical causes of ill-conditioning. Equation (3.37) states that the condition number is the ratio of the largest eigenvalue of K to the smallest. Recalling that K is a generalized stiffness matrix, the smallest eigenvalue corresponds to the stiffness of the softest mode of deformation of the mesh, while the largest eigenvalue is generally well approximated by the stiffness of the stiffest mode of deformation of the stiffest element. This physical interpretation permits some feeling for the condition number of K to be obtained.
One of the more common sources of ill-conditioning in solid mechanics is the use ofsmall elements. Physically, as the typical element size is reduced (i.e.
asthe numberof elements in themeshisincreased), the stiffnessof the individual elements increases. This means that the largest eigenvalue of the mesh will also increase. However, the smallest eigenvalue will remain largely unchanged. Thus the condition number of the stiffness matrix will increase. This dependence on the element size has been treated more mathematically in [2,49].
Conditionsof near incompressibility in three dimensional and plane strain linear elasticity also give rise to ill-conditioned solid mechanics problems. As the material becomes more incompressible, the lowest eigenvalue of K may not change substantially; however, the stiffness of the expansion mode of deformation of the individual elements will increase dramatically (see section 2.3). Another form ofill-conditioning insolid mechanics arises when the problemto be analyzed consists ofmaterials that havewidely different stiffnesses. This case includesthat ofa rigid inclusion.
The problem of ill-conditioning is severe in the area of structural mechan
ics. Finite element meshes consisting of beams produce ill-conditioned matrix equations due to the large stiffness of the axial mode of deformation of the el
ements and the relatively lower stiffnesses of, say, the swaying mode of a frame structure. A similar effect is seen with meshes comprised of shell elements.
When the multigrid method is used to solve ill-conditioned problems in solid mechanics in later chapters of this thesis, its performance will be assessed by usingthe physical interpretation of ill-conditioning described in this section.
Physical reasoning will also be used to explain the performance of the method
when it is applied to problems that are not ill-conditioned in the conventional sense.
Table 3.1: Some of the Linear Stationary Methods of the First Degree
κ
ll
κ12 οκ14
ο[_Ο^ ο ο ο
K22 *23 *24 Ο Κ
26
[_Ο Ο Ο*33 Κ34 κ35 *36 5⅞ ο
«44 *45 *46 Ο K48⅞^
*55 *56 *57 *58 θ j
*66 *67 *68 θ
J
*77 *78 *79∣
*88 *89|
*89}
κ,,
k 12 ∞ jk 14 ∣ 0 0 0
00
*22 *23 *24∣-0JK26^1 0
0 0
*33 *34 *35 *3δj
0 0 0
*44 *45 *4β! θ ! *48∣ θ 1—1 I
BANDED
SKYLINE
*55 *56 *57 *58∣ 0
*66 *67 *68>0
»— —«η
*77 *78 *79∣
*88 *891
*99∣
Figure 3.1: Banded and Skyline Storage Schemes for a Symmetric Matrix.
Chapter 4
The Multigrid Method for Solving Linear Matrix Equations
4.1
Introduction and Historical Background
In this chapter, a multigrid method is described that can be used to solve the linear matrix equation (1.1). Before proceeding with a detailed discussion of the algorithm, it is interesting to briefly comment on its development and previous applications. More detailed historical accounts can be found in [27,56]. The de
velopment ofmultigrid methods can be regarded as part of the wider quest to find efficient iterative algorithms. Southwell [51,52] was one of the first researchers to attempt to improve the basic Jacobi and Gauss-Seidel iteration (or relaxation) schemes; special block and group relaxation methods were developed which in
volved simultaneously relaxing small groups of unknowns. Fedorenko [20] and Bachvalov [3] were the first authors to propose a true multigrid algorithm that explicitly incorporated a coarse grid. However, it was not until Brandt published his extensive paper [12] that the true efficiency of the method was understood.
Brandt’s important contributions include the introduction ofmultigrid methods
for nonlinear problems, adaptive techniques, and the theoretical tool of local Fourier analysis. Since then, the numbers of papers, conferences, and books on the general subject has been growing at an increasing rate.
For the purposes of this discussion, the current multigrid literature can be separated into two broad groups: theoretical development and practical applica tions. Theoretical work, such as [27], has resulted in rather intricate mathematical theories concerning the general convergence of the method. These studies have generally examined the behavior ofthe algorithm whenit is applied to the finite difference solution ofelliptic equations in rectangular domains. The applications of these results to the field of solid and structural mechanics is limited. However, the subsequent algorithm development, together with the main result that the computational work required to solve a given problem is only linearly propor tional to the number of unknowns, are sufficient to motivate research into the application of the multigrid method in this area.
Most of the published applications of the multigrid method arise from the fieldof fluid mechanics. In particular, potential and Euler flows, as well as prob
lems governed by the full Navier-Stokes equations, have been solved [13,17,48]. However, the applications dealing explicitly with solid and structural mechanics problems are conspicuous by their absence. The only paper at present known to the author is [5], in which some fairly small (about 2,000 degrees-of-freedom) two dimensional plane stress, plane strain, and axisymmetric linear elastic problems were solved using the finite element method. Other authors have also briefly mentioned the application of the multigrid method to the solution of solid and structural mechanics problems. An applicationto membrane and plate problems
is given in [11], as well as a discussion of the effect of lockingon the performance of the method. The eigenvalues and eigenvectors of a rectangular plate were obtained using the multigrid method in [24].
There is still a great deal of work to be done on the application of the multigrid method to the solution of solid and structural mechanics problems of engineeringinterest. The remainder ofthischapterwilldescribein detaila multi- grid algorithm that can be used to solve the linear matrix equation (1.1) that arises from the finite element discretization ofa linear solid mechanics problem.