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Numerical Finite and Boundary Element Methods
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Numerical Finite and Boundary Element Methods
Reviewing the previous chapters would indicate that analytical solutions to elasticity problems are normally accomplished for regions and loadings with relatively simple geometry. For example, many solutions can be developed for two-dimensional problems, while only a limited number exist for three dimensions. Solutions are commonly available for problems with simple shapes such as those having boundaries coinciding with Cartesian, cylindrical, and spherical coordinate surfaces. Unfortunately, problems with more general boundary shape and loading are commonly intractable or require very extensive mathematical analysis and numerical evaluation. Because most real-world problems involve structures with complicated shape and loading, a gap exists between what is needed in applications and what can be solved by analytical closed-form methods. Over the years, this need to determine deformation and stresses in complex problems has lead to the development of many approximate and numerical solution methods (see brief discussion in Section 5.7). Approximate methods based on energy techniques were outlined in Section 6.7, but it was pointed out that these schemes have limited success in developing solutions for problems of complex shape. Methods of numerical stress analysis normally recast the mathematical elasticity boundary value problem into a direct numerical routine. One such early scheme is the finite difference method (FDM) in which derivatives of the governing field equations are replaced by algebraic difference equations. This method generates a system of algebraic equations at various computational grid points in the body, and the solution to the system determines the unknown variable at each grid point. Although simple in concept, FDM has not been able to provide a useful and accurate scheme to handle general problems with geometric and loading complexity. Over the past few decades, two methods have emerged that provide necessary accuracy, general applicability, and ease of use. This has led to their acceptance by the stress analysis community and has resulted in the development of many private and commercial computer codes implementing each numerical scheme. The first of these techniques is known as the finite element method (FEM) and involves dividing the body under study into a number of pieces or subdomains called elements. The solution is then approximated over each element and is quantified in terms of values at special locations within the element called the nodes. The discretization process establishes an
473
Numerical Finite and Boundary Element Methods
Reviewing the previous chapters would indicate that analytical solutions to elasticity problems are normally accomplished for regions and loadings with relatively simple geometry. For example, many solutions can be developed for two-dimensional problems, while only a limited number exist for three dimensions. Solutions are commonly available for problems with simple shapes such as those having boundaries coinciding with Cartesian, cylindrical, and spherical coordinate surfaces. Unfortunately, problems with more general boundary shape and loading are commonly intractable or require very extensive mathematical analysis and numerical evaluation. Because most real-world problems involve structures with complicated shape and loading, a gap exists between what is needed in applications and what can be solved by analytical closed-form methods. Over the years, this need to determine deformation and stresses in complex problems has lead to the development of many approximate and numerical solution methods (see brief discussion in Section 5.7). Approximate methods based on energy techniques were outlined in Section 6.7, but it was pointed out that these schemes have limited success in developing solutions for problems of complex shape. Methods of numerical stress analysis normally recast the mathematical elasticity boundary value problem into a direct numerical routine. One such early scheme is the finite difference method (FDM) in which derivatives of the governing field equations are replaced by algebraic difference equations. This method generates a system of algebraic equations at various computational grid points in the body, and the solution to the system determines the unknown variable at each grid point. Although simple in concept, FDM has not been able to provide a useful and accurate scheme to handle general problems with geometric and loading complexity. Over the past few decades, two methods have emerged that provide necessary accuracy, general applicability, and ease of use. This has led to their acceptance by the stress analysis community and has resulted in the development of many private and commercial computer codes implementing each numerical scheme. The first of these techniques is known as the finite element method (FEM) and involves dividing the body under study into a number of pieces or subdomains called elements. The solution is then approximated over each element and is quantified in terms of values at special locations within the element called the nodes. The discretization process establishes an
473