Object-Oriented Finite Element Analysis by Graham Charles Archer B.A.Sc. (University of Waterloo) 1985 M.A.Sc. (University of Waterloo) 1986 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering-Civil Engineering in the GRADUATE DIVISION of the UNIVERSITY of CALIFORNIA at BERKELEY Committee in charge: Professor Christopher Thewalt, Chair Professor Gregory Fenves Professor James Demmel 1996 The dissertation of Graham Charles Archer is approved: University of California at Berkeley 1996 Object-Oriented Finite Element Analysis Copyright 1996 by Graham Charles Archer Abstract Object-Oriented Finite Element Analysis by Graham Charles Archer Doctor of Philosophy in Civil Engineering University of California at Berkeley Professor Christopher Thewalt, Chair Over the last 30 years, the finite element method has gained wide acceptance as a general purpose tool for structural modeling and simulation. Typical finite element programs consist of several hundred thousand lines of procedural code, usually written in FORTRAN. The codes contain many complex data structures which are accessed throughout the program. For the sake of efficiency, the various components of the program often take advantage of this accessibility. For example, elements, nodes, and constraints may directly access the matrices and vectors involved in the analysis to obtain and transmit their state information. Thus they become intimately tied to the analysis data structures. This not only compounds the complexity of the components, by requiring knowledge of the data structures, but writers of new analysis procedures must be aware of all components that access the data. Modification or extension of a portion of the code requires a high degree of knowledge of the entire program. Solution procedures and modeling techniques become hard coded. The resulting code is inflexible and presents a barrier to practicing engineers and researchers. 1 Recoding these systems in a new language will not remove this inflexibility. Instead, a redesign using an object-oriented philosophy is needed. This abstraction forms a stable definition of objects in which the relationships between the objects are explicitly defined. The implicit reliance on another component's data does not occur. Thus, the design can be extended with minimal effort. The application of object-oriented design to the finite element method has several advantages. The primary advantage is that it encourages the developer to abstract out the essential immutable qualities of the components of the finite element method. This abstraction forms the definition of objects that become the building blocks of the software. The class definitions encapsulate both the data and operations on the data, and provide an enforceable interface by which other components of the software may communicate with the object. Once specified, the object interfaces are frozen. It is possible to extend the interface later without affecting other code, but it should be noted that modifying the existing elements of an interface would require changes throughout the rest of the program wherever the interface was used. The internal class details that are needed to provide the desired interface are invisible to the rest of the program and, therefore, these implementation details may be modified without affecting other code. Thus, the design forms a stable base that can be extended with minimum effort to suit a new task. Due to the encapsulation enforcement inherent in object- oriented languages, new code will work seamlessly with old code. In fact old code may call new code. The system design and a prototype implementation for the finite element method is presented. The design describes the abstraction for each class and specifies the interface for the abstraction. The implementation of a class, and the interaction between 2 objects must conform to the interface definition. Furthermore, recognizing that most finite element development involves the addition of elements, new solution strategies, or new matrix storage schemes, special care was taken to make these interfaces as flexible as possible. The flexibility is provided by an object that is responsible for automatically and efficiently transforming between coordinate systems to relieve element and solution writers from this task. Elements authors are free to work in any convenient coordinate system rather than requiring them to use the degrees-of-freedom described by the modeler. Similarly, the writers of new solution strategies need not deal with any model specific details or transformations, but rather are presented only with the equations to be solved. The transformation from element coordinate systems to final equations, including the effects of constraints and equation reordering, is transparently handled without element or solution writer intervention. This separation of tasks also allows analysis objects to use any matrix storage scheme that is convenient for that particular method. 3 Table of Contents List of Figures.........................................................................................................viii List of Tables..........................................................................................................xi 1. Introduction.......................................................................................................1 1.1 Problem.................................................................................................1 1.2 Objective..............................................................................................3 1.3 Object-Oriented Philosophy as a Solution..............................................4 1.4 Organization.........................................................................................5 2. Literature Review...............................................................................................7 2.1 G. R. Miller, et al...................................................................................10 2.2 T. Zimmermann, et al............................................................................11 2.3 Jun Lu, et al...........................................................................................12 2.4 J. W. Baugh, et al...................................................................................14 2.5 H. Adeli, et al.........................................................................................15 2.6 Discussion..............................................................................................16 3. Methodologies....................................................................................................19 3.1 Design Formalisms.................................................................................20 3.1.1 Design Intent...........................................................................20 3.1.2 Entity Relationship Diagrams...................................................20 3.1.3 Responsibility Description.......................................................21 3.1.4 Event Flow Diagram................................................................22 3.1.5 Interface Definition Table........................................................23 3.2 Implementation in C++..........................................................................24 3.2.1 Associations............................................................................25 iii 3.2.2 Inheritance...............................................................................26 3.2.3 Interfaces.................................................................................27 3.2.4 Libraries..................................................................................28 3.2.4.1 Container Classes......................................................29 3.2.4.2 Numerical Object Library..........................................30 3.3 Implementation Formalisms....................................................................30 3.3.1 C++ Interfaces.........................................................................31 3.3.2 Demonstration Code................................................................32 4. Object Model Design..........................................................................................35 4.1 Top Level Objects..................................................................................38 4.1.1 Model......................................................................................40 4.1.2 Map.........................................................................................42 4.1.3 Analysis...................................................................................48 4.2 Components of the Model......................................................................52 4.2.1 Degree of Freedom Group (Node)...........................................52 4.2.2 Element...................................................................................59 4.2.3 Material, Action/Deformation, and Constitutive Model............65 4.2.4 Constraint................................................................................70 4.2.5 Displacement Boundary Condition...........................................73 4.2.6 Load Case...............................................................................75 4.3 Handlers................................................................................................82 4.3.1 Constraint Handler..................................................................83 4.3.2 Reorder Handler......................................................................86 4.3.3 Matrix Handler........................................................................89 4.4 Utility Objects........................................................................................91 4.4.1 Geometry................................................................................91 iv 4.4.2 Numerical Objects...................................................................95 4.4.3 Augmented Vector and Matrix Objects....................................98 5. Behavior of High Level Classes..........................................................................104 5.1 Element.................................................................................................106 5.1.1 Element Geometry...................................................................106 5.1.2 Communication of Element Properties.....................................107 5.1.3 Element State..........................................................................109 5.1.4 Action/Deformation.................................................................110 5.1.5 Resisting Force........................................................................111 5.2 Constitutive Model................................................................................111 5.2.1 Material Data..........................................................................112 5.2.2 Action/Deformation.................................................................113 5.2.3 Constitutive Model State.........................................................115 5.3 Analysis.................................................................................................116 5.3.1 Analysis Constructors..............................................................117 5.3.2 Analysis State..........................................................................119 5.3.3 Assembly.................................................................................120 5.3.4 Treatment of Boundary Conditions..........................................121 5.4 Constraint Handler.................................................................................124 5.4.1 Transformation Theory............................................................124 5.4.2 Selection of Analysis Unknowns..............................................126 5.4.3 Data Types.............................................................................127 5.4.4 Implementation.......................................................................128 5.5 Map.......................................................................................................133 5.5.1 Example..................................................................................133 5.5.1.1 Transformation of a Stiffness Matrix.........................135 v 5.5.1.2 Processing of Responses..........................................138 5.5.2 Implementation.......................................................................139 6. Extensions and Examples.....................................................................................147 6.1 Examples of Analysis.............................................................................148 6.1.1 Linear Static Analysis..............................................................149 6.1.2 Nonlinear Static Event to Event Analysis.................................151 6.1.3 Nonlinear Static Analysis with Newton-Raphson Iteration.......155 6.1.4 Nonlinear Dynamic Analysis with Newton-Raphson Iteration...........................................................................................157 6.2 Substructuring......................................................................................163 6.2.1 FeaSuperElement.....................................................................167 6.2.2 FeaSuperElementLoad.............................................................179 6.2.3 FeaSuperElementInitialState....................................................180 6.3 2D Beam Element With End Hinges......................................................181 6.3.1 Moment-Rotation Elastic-Perfectly Plastic Constitutive Model..............................................................................................182 6.3.2 2D Rectangular Coordinate System........................................191 6.3.3 2D Beam Element With End Hinges.......................................196 6.4 Building a Model...................................................................................215 6.5 Using the Finite Element Program..........................................................220 6.5.1 Linear Static Analysis..............................................................221 6.5.2 Nonlinear Static Analysis.........................................................222 6.5.3 Nonlinear Dynamic Analysis....................................................225 7. Conclusions and Future Work.............................................................................229 7.1 Conclusions...........................................................................................229 7.1.1 Architecture............................................................................230 vi