Evaluation Of Stiffness Matrices Research Articles

Sparse Generalized Multiscale Finite Element Methods and their applications

In a number of previous papers, local (coarse grid) multiscale model reduction techniques are developed using a Generalized Multiscale Finite Element Method. In these approaches, multiscale basis functions are constructed using local snapshot spaces, where a snapshot space is a large space that represents the solution behavior in a coarse block. In a number of applications (e.g., those discussed in the paper), one may have a sparsity in the snapshot space for an appropriate choice of a snapshot space. More precisely, the solution may only involve a portion of the snapshot space. In this case, one can use sparsity techniques to identify multiscale basis functions. In this paper, we consider two such sparse local multiscale model reduction approaches. In the first approach (which is used for parameter-dependent multiscale PDEs), we use local minimization techniques, such as sparse POD, to identify multiscale basis functions, which are sparse in the snapshot space. These minimization techniques use l1 minimization to find local multiscale basis functions, which are further used for finding the solution. In the second approach (which is used for the Helmholtz equation), we directly apply l1 minimization techniques to solve the underlying PDEs. This approach is more expensive as it involves a large snapshot space; however, in this example, we cannot identify a local minimization principle, such as local generalized SVD. All our numerical results assume the sparsity and we discuss this assumption for the snapshot spaces. Moreover, we discuss the computational savings provided by our approach. The sparse solution allows a fast evaluation of stiffness matrices and downscaling the solution to the fine grid since the reduced dimensional solution representation is sparse in terms of local snapshot vectors. Numerical results are presented, which show the convergence of the proposed method and the sparsity of the solution.

An Introduction to Linear and Nonlinear Finite Element Analysis: A Computational Approach

9R1. An Introduction to Linear and Nonlinear Finite Element Analysis: A Computational Approach. - Edited by PK Kythe and Dongming Wei (Dept of Math, Univ of New Orleans, New Orleans LA 70148-0001). Birkhauser, Boston. 2004. 445 pp. ISBN 0-8176-4308-7. $79.95.Reviewed by M Okrouhlik (Dept of Solids, Inst of Thermomech, Acad of Sci, Dolejskova 5, 182 00 Prague 8, Czech Republic).The book is intended as a textbook for undergraduate and graduate students from engineering, geophysics, and applied mathematics. Very broad spectrum of engineering problems is covered—the examples are taken from structural, mechanical, electrical, and chemical fields. Generally, the book deals with linear and nonlinear problems in radiation, heat transfer, mechanics of elastic and plastic media, continuum mechanics, non-Newtonian fluid flows, and electromagnetics. The book is composed of 14 chapters, 6 appendices, a bibliography, and subject index. Chapter 1 deals with weak variational formulation of a boundary value problem, Galerkin and Rayleigh-Ritz weighted residual methods. Chapter 2 presents one-dimensional local and global interpolation functions. Chapter 3 explains the Galerkin method and applies it to a one-dimensional second-order equation using liner and quadratic elements. Chapter 4 treats one-dimensional fourth-order equation (beam). Chapter 5 introduces linear triangular and bilinear four-node rectangular elements. Chapter 6 is devoted to two-dimensional problems with a single scalar variable. Stiffness matrix and load vector are derived. In Chapter 7 the two-dimensional boundary value problems as heat exchange, torsion, and seepage are treated. Chapter 8 deals with axisymmetric linear and nonlinear heat transfer problems in solid and fluids. Chapter 9 is devoted to transient problems and numerical time integration. Chapter 10 treats nonlinear problems in one dimension—radiation heat transfer, stress in elastoplastic bars, non-Newtonian fluid flow in between parallel plates, and turbulent flows in tubes. For the numerical solution the Newton method, the method of the steepest descent and conjugate gradient methods are used. Chapter 11 presents the steady-state problems of plane elasticity. Linear triangular and bilinear rectangular elements are treated; stiffness matrices and load vectors are derived. Assembling is clearly described but implementation considerations are not considered. Chapter 12 introduces the penalty method. It is applied to the treatment of both Newtonian and power law non-Newtonian Stokes flow. Chapter 13 deals with vibration analysis—elastic rods, Euler beams, and in-plane vibration of an elastic plate are treated. The last chapter contains the computer codes in Mathematica, Matlab, and Fortran. They form a vivid complement to selected problems appearing in the book. Results of computer runs are presented in the tabular form. There are six appendices, labeled A to G, in the book. They contain useful complementary information and are subsequently devoted to overview of classical integration formulas, evaluation of stiffness matrices for triangular and rectangular elements for a chosen geometry, time-step marching algorithms (forward and backward difference schemes, Cranck-Nicolson formulas, Newmark method, etc). Also the concept of isoparametric elements is briefly explained, Green identities are shown and Gauss quadrature formulas are derived. In the last appendix the classical minimization methods (method of the steepest gradient and conjugate gradient method) are presented. There are 87 examples and 148 exercises and 152 figures in the book. Solving the problems both SI and imperial units are used. The finite element analysis is clearly presented with rigorous mathematical treatment of its background and accompanied by numerous examples. Finite element methodology is covered briefly being saliently combined with a proper computer implementation. Programs and subroutines written in Fortran, Matlab, and Mathematica, treating the examples presented in the book, form a consistent part of the book. The book is carefully edited and printed. The only misprint I found is on p. 403 where there is a missing zero term in a matrix. From the reviewer’s point of view the book has high educational value stemming from the fact that a very large spectrum of engineering topics is covered. As such the book might well be a good purchase both for university libraries and individuals.

Evaluation of element stiffness matrices on CDC STAR-100 computer

An algorithm is presented for the evaluation of the stiffness matrices of higher-order elements on the CDC STAR-100 computer. Discussion is focused on the organization of the computation and the mode of storage of the different arrays to take advantage of the STAR pipeline (streaming) capability. An assessment is made of the performance of the proposed algorithm for generating the stiffness matrices of two higher order composite shallow shell and plate elements having 80 and 176 degrees of freedom. Also, estimates are given of the CPU times required to evaluate the stiffness coefficients of three-dimensional hexahedral elements using the proposed algorithm.

Comparison and evaluation of stiffness matrices.

Previous developments in the comparison and evaluation of stiffness matrices are reviewed. A simplification to the method of making comparisons on the basis of strain energy is presented, and it is shown that the results of element stiffness matrix comparisons apply to the structure. The theoretical basis for obtaining approximate qualitative comparisons from stiffness matrix eigenvalues is described, and the existing hypothesis on their use evaluated. Some plane stress stiffness matrices for square, isotropic elements are compared, and it is shown that it is reasonable to expect bounding of the strain energy by varying a stiffness matrix parameter. A method of comparing stiffness matrices of different orders is proposed. It also is shown that square elements generally will provide a better approximation to the strain energy than constant-stress triangular elements.