Finite Element Analysis - Thermomechanics of Solids Part 8 pps

117 Introduction to the Finite- Element Method 8.1 INTRODUCTION In thermomechanical members and structures, finite-element analysis (FEA) is typi- cally invoked to compute displacement and temperature fields from known applied loads and heat fluxes. FEA has emerged in recent years as an essential resource for mechanical and structural designers. Its use is often mandated by standards such as the ASME Pressure Vessel Code, by insurance requirements, and even by law. Its acceptance has benefited from rapid progress in related computer hardware and soft- ware, especially computer-aided design (CAD) systems. Today, a number of highly developed, user-friendly finite-element codes are available commercially. The purpose of this chapter is to introduce finite-element theory and practice. The next three chapters focus on linear elasticity and thermal response, both static and dynamic, of basic structural members. After that, nonlinear thermomechanical response is considered. In FEA practice, a design file developed using CAD is often “imported” into finite- element codes, from which point little or no additional effort is required to develop the finite-element model and perform sophisticated thermomechanical analysis and simulation. CAD integrated with an analysis tool, such as FEA, is an example of computer-aided engineering (CAE). CAE is a powerful resource with the potential of identifying design problems much more efficiently and rapidly than by “trial and error.” A major FEM application is the determination of stresses and temperatures in a component or member in locations where failure is thought most likely. If the stresses or temperatures exceed allowable or safe values, the product can be rede- signed and then reanalyzed. Analysis can be diagnostic, supporting interpretation of product-failure data. Analysis also can be used to assess performance, for example, by determining whether the design-stiffness coefficient for a rubber spring is attained. 8.2 OVERVIEW OF THE FINITE-ELEMENT METHOD Consider a thermoelastic body with force and heat applied to its exterior boundary. The finite-element method serves to determine the displacement vector u ( X , t ) and the temperature T( X , t ) as functions of the undeformed position X and time t . The process of creating a finite-element model to support the design of a mechanical system can be viewed as having (at least) eight steps: 1. The body is first discretized, i.e., it is modeled as a mesh of finite elements connected at nodes. 2. Within each element, interpolation models are introduced to provide approximate expressions for the unknowns, typically u ( X , t ) and T( X , t ), 8 0749_Frame_C08 Page 117 Wednesday, February 19, 2003 5:08 PM © 2003 by CRC CRC Press LLC 118 Finite Element Analysis: Thermomechanics of Solids in terms of their nodal values, which now become the unknowns in the finite-element model. 3. The strain-displacement relation and its thermal analog are applied to the approximations for u and T to furnish approximations for the (Lagrangian) strain and the thermal gradient. 4. The stress-strain relation and its thermal analog (Fourier’s Law) are applied to obtain approximations to stress S and heat flux q in terms of the nodal values of u and T. 5. Equilibrium principles in variational form are applied using the various approx- imations within each element, leading to element equilibrium equations . 6. The element equilibrium equations are assembled to provide a global equilibrium equation . 7. Prescribed kinematic and temperature conditions on the boundaries ( con- straints ) are applied to the global equilibrium equations, thereby reducing the number of degrees of freedom and eliminating “rigid-body” modes. 8. The resulting global equilibrium equations are then solved using computer algorithms. The output is postprocessed. Initially, the output should be compared to data or benchmarks, or otherwise validated, to establish that the model correctly repre- sents the underlying mechanical system. If not satisfied, the analyst can revise the finite-element model and repeat the computations. When the model is validated, postprocessing, with heavy reliance on graphics, then serves to interpret the results, for example, determining whether the underlying design is satisfactory. If problems with the design are identified, the analyst can then choose to revise the design. The revised design is modeled, and the process of validation and interpretation is repeated. 8.3 MESH DEVELOPMENT Finite-element simulation has classically been viewed as having three stages: pre- processing, analysis , and postprocessing . The input file developed at the preprocess- ing stage consists of several elements: 1. control information (type of analysis, etc.) 2. material properties (e.g., elastic modulus) 3. mesh (element types, nodal coordinates, connectivities) 4. applied force and heat flux data 5. supports and constraints (e.g., prescribed displacements) 6. initial conditions (dynamic problems) In problems without severe stress concentrations, much of the mesh data can be developed conveniently using automatic-mesh generation. With the input file devel- oped, the analysis processor is activated and “raw” output files are generated. The postprocessor module typically contains (interfaces to) graphical utilities, thus facil- itating display of output in the form chosen by the analyst, for example, contours of the Von Mises stress. Two problems arise at this stage: validation and interpretation . 0749_Frame_C08 Page 118 Wednesday, February 19, 2003 5:08 PM © 2003 by CRC CRC Press LLC Introduction to the Finite-Element Method 119 The analyst can use benchmark solutions, special cases, or experimental data to validate the analysis. With validation, the analyst gains confidence in, for example, the mesh. He or she still may face problems of interpretation, particularly if the output is voluminous. Fortunately, current graphical-display systems make interpre- tation easier and more reliable, such as by displaying high stress regions in vivid colors. Postprocessors often allow the analyst to “zoom in” on regions of high interest, for example, where rubber is highly confined. More recent methods based on virtual-reality technology enable the analyst to fly through and otherwise become immersed in the model. The goal of mesh design is to select the number and location of finite-element nodes and element types so that the associated analyses are sufficiently accurate. Several methods include automatic-mesh generation with adaptive capabilities, which serve to produce and iteratively refine the mesh based on a user-selected error tolerance. Even so, satisfactory meshes are not necessarily obtained, so that model editing by the analyst may be necessary. Several practical rules are as follows: 1. Nodes should be located where concentrated loads and heat fluxes are applied. 2. Nodes should be located where displacements and temperatures are con- strained or prescribed in a concentrated manner, for example, where “pins” prevent movement. 3. Nodes should be located where concentrated springs and masses and their thermal analogs are present. 4. Nodes should be located along lines and surface patches, over which pressures, shear stresses, compliant foundations, distributed heat fluxes, and surface convection are applied. 5. Nodes should be located at boundary points where the applied tractions and heat fluxes experience discontinuities. 6. Nodes should be located along lines of symmetry. 7. Nodes should be located along interfaces between different materials or components. 8. Element-aspect ratios (ratio of largest to smallest element dimensions) should be no greater than, for example, five. 9. Symmetric configurations should have symmetric meshes. 10. The density of elements should be greater in domains with higher gradi- ents. 11. Interior angles in elements should not be excessively acute or obtuse, for example, less than 45 ° or greater than 135 ° . 12. Element-density variations should be gradual rather than abrupt. 13. Meshes should be uniform in subdomains with low gradients. 14. Element orientations should be staggered to prevent “bias.” In modeling a configuration, a good practice is initially to develop the mesh locally in domains expected to have high gradients, and thereafter to develop the mesh in the intervening low-gradient domains, thereby “reconciling” the high-gradient domains. 0749_Frame_C08 Page 119 Wednesday, February 19, 2003 5:08 PM © 2003 by CRC CRC Press LLC 120 Finite Element Analysis: Thermomechanics of Solids There are two classes of errors in finite-element analysis: Modeling error ensues from inaccuracies in such input data as the material properties, boundary conditions, and initial values. In addition, there often are compromises in the mesh, for example, modeling sharp corners as rounded. Numerical error is primarily due to truncation and round-off. As a practical matter, error in a finite-element simulation is often assessed by comparing solutions from two meshes, the second of which is a refinement of the first. The sensitivity of finite-element computations to error is to some extent con- trollable. If the condition number of the stiffness matrix (the ratio of the maximum to the minimum eigenvalue) is modest, sensitivity is reduced. Typically, the condition number increases rapidly as the number of nodes in a system grows. In addition, highly irregular meshes tend to produce high-condition numbers. Models mixing soft components, for example, rubber, with stiff components, such as steel plates, are also likely to have high-condition numbers. Where possible, the model should be designed to reduce the condition number. 0749_Frame_C08 Page 120 Wednesday, February 19, 2003 5:08 PM © 2003 by CRC CRC Press LLC . , t ), 8 0749_Frame_C 08 Page 117 Wednesday, February 19, 2003 5: 08 PM © 2003 by CRC CRC Press LLC 1 18 Finite Element Analysis: Thermomechanics of Solids in terms of their nodal. hardware and soft- ware, especially computer-aided design (CAD) systems. Today, a number of highly developed, user-friendly finite -element codes are available commercially. The purpose of this chapter. 117 Introduction to the Finite- Element Method 8. 1 INTRODUCTION In thermomechanical members and structures, finite -element analysis (FEA) is typi- cally invoked to compute displacement