Belytschko T. - Introduction

T. Belytschko, Introduction, December 16, 1998CHAPTER 1INTRODUCTIONby Ted BelytschkoNorthwestern UniversityCopyright 19961.1 NONLINEAR FINITE ELEMENTS IN DESIGNNonlinear finite element analysis is an essential component of computeraided design. Testing of prototypes is increasingly being replaced by simulationwith nonlinear finite element methods because this provides a more rapid and lessexpensive way to evaluate design concepts and design details. For example, inthe field of automotive design, simulation of crashes is replacing full scale tests,both for the evaluation of early design concepts and details of the final design,such as accelerometer placement for airbag deployment, padding of the interior,and selection of materials and component cross-sections for meetingcrashworthiness criteria.

In many fields of manufacturing, simulation is speedingthe design process by allowing simulation of processes such as sheet-metalforming, extrusion of parts, and casting. In the electronics industries, simulationis replacing drop-tests for the evaluation of product durability.For both users and developers of nonlinear finite element programs, anunderstanding of the fundamental concepts of nonlinear finite element analysis isessential. Without an understanding of the fundamentals, a user must treat thefinite element program as a black box that provides simulations.

However, evenmore so than linear finite element analysis, nonlinear finite element analysisconfronts the user with many choices and pitfalls. Without an understanding ofthe implication and meaning of these choices and difficulties, a user is at a severedisadvantage.The purpose of this book is to describe the methods of nonlinear finiteelement analysis for solid mechanics. Our intent is to provide an integratedtreatment so that the reader can gain an understanding of the fundamentalmethods, a feeling for the comparative usefulness of different approaches and anappreciation of the difficulties which lurk in the nonlinear world. At the sametime, enough detail about the implementation of various techniques is given sothat they can be programmed.Nonlinear analysis consists of the following steps:1. development of a model;2. formulation of the governing equations;3.

discretization of the equations;4. solution of the equations;1-1T. Belytschko, Introduction, December 16, 19985. interpretation of the results.Modeling is a term that tends to be used for two distinct tasks inengineering. The older definition emphasizes the extraction of the essentialelements of mechanical behavior. The objective in this approach is to identify thesimplest model which can replicate the behavior of interest. In this approach,model development is the process of identifying the ingredients of the modelwhich can provide the qualitative and quantitative predictions.A second approach to modeling, which is becoming more common inindustry, is to develop a detailed, single model of a design and to use it toexamine all of the engineering criteria which are of interest.

The impetus for thisapproach to modeling is that it costs far more to make a model or mesh for anengineering product than can be saved through reduction of the model byspecializing it for each application. For example, the same finite element modelof a laptop computer can be used for a drop-test simulation, a linear static analysisand a thermal analysis. By using the same model for all of these analyses, asignificant amount of engineering time can be saved. While this approach is notrecommended in all situations, it is becoming commonplace in industry.

In thenear future the finite element model may serve as a prototype that can be used forchecking many aspects of a design’s performance. The decreasing cost ofcomputer time and the increasing speed of computers make this approach highlycost-effective. However the user of finite element software must still able toevaluate the suitability of a model for a particular analysis and understand itslimitations.The formulation of the governing equations and their discretization islargely in the hands of the software developers today. However, a user who doesnot understand the fundamentals of the software faces many perils, for someapproaches and software may be unsuitable. Furthermore, to convertexperimental data to input, the user must be aware of the stress and strainmeasures used in the program and by the experimentalist who provided materialdata.

The user must understand the sensitivity of response to the data and how toasses it. An effective user must be aware of the likely sources of error, how tocheck for these errors and estimate their magnitudes, and the limitations andstrengths of various algorithms.The solution of the discrete equations also presents a user with manychoices.

An inappropriate choice will result in very long run-times which canprevent him from obtaining the results within the time schedule. Anunderstanding of the advantages and disadvantages and the approximate computertime required for various solution procedures are invaluable in the selection of agood strategy for developing a reasonable model and selecting the solutionprocedure.The user’s role is most crucial in the interpretation of results. In additionto the approximations inherent even in linear finite element models, nonlinearanalyses are often sensitive to many factors that can make a single simulationquite misleading.

Nonlinear solids can undergo instabilities, their response can besensitive to imperfections, and the results can depend dramatically on materialparameters. Unless the user is aware of these phenomena, the possibility of amisinterpretation of simulation results is quite possible.1-2T. Belytschko, Introduction, December 16, 1998In spite of these many pitfalls, our views on the usefulness and potentialof nonlinear finite element analyses are very sanguine. In many industries,nonlinear finite element analysis have shortened design cycles and dramaticallyreduced the need for prototype tests. Simulations, because of the wide variety ofoutput they produce and the ease of doing what-ifs, can lead to tremendousimprovements of the engineer's understanding of the basic physics of a product'sbehavior under various environments.

While tests give the gross but importantresult of whether the product withstands a certain environment, they usuallyprovide little of the detail of the behavior of the product on which a redesign canbe based if the product does not meet a test. Computer simulations, on the otherhand, give detailed histories of stress and strain and other state variables, which inthe hands of a good engineer give valuable insight into how to redesign theproduct .Like many finite element books, this book presents a large variety ofmethods and recipes for the solution of engineering and scientific problems by thefinite element method.

However, in order to preserve a pedagogic character, wehave interwoven several themes into the book which we feel are of centralimportance in nonlinear analysis. These include the following:1. the selection of appropriate methods for the problem at hand;2. the selection of a suitable mesh description and kinematic and kineticdescriptions for a given problem;3.

the examination of stability of the solution and the solution procedure;4. an awareness of the smoothness of the response of the model and itsimplication on the quality and cost of the solution;5. the role of major assumptions and the likely sources of error.The selection of an appropriate mesh description, i.e. whether aLagrangian, Eulerian or arbitrary Lagrangian Eulerian mesh is used, is veryimportant for many of the large deformation problems encountered in processsimulation and failure analysis. The effects of mesh distortion need to beunderstood, and the advantages of different types of mesh descriptions should beborne in mind in the selection.

There are many situations where a continuousremeshing or arbitrary Lagrangian Eulerian description is most suitable.The issue of the stability of solution is central in the simulation ofnonlinear processes. In numerical simulation, it is possible to obtain solutionswhich are not physically stable and therefore quite meaningless. Many solutionsare sensitive to imperfections or material and load parameters; in some cases,there is even sensitivity to the mesh employed in the solution. A knowledgeableuser of nonlinear finite element software must be aware of these characteristicsand the associated pitfalls. Otherwise the results obtained by elaborate computersimulations can be quite misleading and lead to incorrect design decisions.The issue of smoothness is also ubiquitous in nonlinear finite elementanalysis.

Lack of smoothness degrades the robustness of most algorithms and canintroduce undesirable noise into the solution. Techniques have been developedwhich improve the smoothness of the response; these are generally calledregularization procedures. However, regularization procedures are often not1-3T. Belytschko, Introduction, December 16, 1998based on physical phenomena and in many cases the constants associated with theregularization are difficult to determine.

Therefore, an analyst is often confrontedwith the dilemma of whether to choose a method which leads to smoothersolutions or to deal with a discontinuous response. An understanding of theeffects of regularization parameters, the presence of hidden regularizations, suchas penalty methods in contact-impact, and an appreciation of the benefits of thesemethods is highly desirable.The accuracy and stability of solutions is a difficult consideration innonlinear analysis. These issues manifest themselves in many ways.