ON THE IMPLEMENTATION OF THE P-ADAPTIVE FINITE ELEMENT METHOD USING THE OBJECT

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1 ON THE IMPLEMENTATION OF THE P-ADAPTIVE FINITE ELEMENT METHOD USING THE OBJECT ORIENTED PROGRAMMING PHILOSOFY Philippe R.B. Devloo Faculdade de Engenharia Civil - UNICAMP Carlos Armando Magalhaes Duarte & Andre Tamagnini Noel Departamento de Engenharia Mec^anica - UFSC phil@fec.unicamp.br SUMMARY In this paper, an implementation of the p-adaptive nite element method is described using the object oriented programming philosofy. It is shown that, using the object oriented programming language C++, the p-adaptive method can be implemented with relative ease. The structure of the resulting program permits the same nite element class to be applied to a variety of computational mechanics problems. The blockoriented banded matrix implementation allows to increase and/or decrease the polynomial order of a given element with high eciency. The concepts oered by the object oriented programming philosofy can be used to extend the nite element program to include new elds of computational mechanics and implement dierent algorithms such as h-p adaptivity, multigrid, etc. The authors hope that their eort will form the basis of future scientic cooperation projects about the development of adaptive nite element software. 1 INTRODUCTION In recent years, the object oriented programming philosofy has established itself as the programming methodology of choice for such areas as operating systems, graphical user interfaces, graphics programming and databases. ([1]). In this work, an object oriented implementation of a nite element program is presented, bringing the benets of object oriented programming to scientic computing. Rather than proposing an object oriented implementation of the traditional nite element data structures, a new structure is developed which is particularly suited for implementing adaptive nite element algorithms. In a parallel eort, an object oriented system-independent graphics environment is being developed which facilitates the development of interactive programs. In the near future, both eorts will be combined to form a p-adaptive nite element program with a graphical interface. The Program Development Cycle During the development of an object oriented program, it is common to work with both a bottom up and top down design. With top down design, we denote the description of the global organization of the program. At this level, the interaction of the dierent classes is described without considering the details of their implementation. In the bottom up design, the ner granularity of the

2 program is discussed. In a typical nite element program, this concerns the implementation of an element, a node, an integration rule, etc. These individual modules then interact to form a complete program. Both aproaches are usually combined to form a functional program. In the rst section, a global description of the nite element program is given. In the second and further sections, individual building blocks are described, motivating the choices which were made when dening their behaviour. 2 GENERAL ORGANIZATION The main purpose of a nite element program is to perform a nite element analysis. Therefore, the top level class in the program is the analysis class. The analysis class combines a datale, a geometric grid, a computable grid, a matrix and a solution vector in order to perform the global steps of a nite element analysis : data initialization, matrix assembly and decomposition. Using these four objects, the analysis class reads a datale and builds the geometric and computable grid. The matrix object stored in the analysis class allows for a choice of various equation solvers. The geometric grid contains the data items needed to describe the geometry of the domain of the problem. As such the geometric grid functions as a database of geometric nodes, geometric elements and geometric descriptions of boundary conditions (elements and sides, nodes, etc). Putting the information in a database like structure facilitates the use of out-of-core memory schemes and save/restore operations. The computable grid contains the data items necessary to dene the dierential equation, its boundary conditions and the nite element interpolation. These items are dened through lists of materials, computable elements, computable nodes and boundary conditions. Like the geometric grid, the computable grid functions as a database of these items rather than as an object which does the actual computations. The nite element operations such a computation of the stiness matrix, assembly and decomposition of the global matrix are done by the analysis class. The matrix class consists of a family of classes which represent the global stiness matrix of the nite element approximation. Depending on the class being used, the matrix object uses a banded, block banded or element by element storage scheme. The implementation of the block banded storage scheme will be discussed in a later section of this paper. Finally, the datale consists of a family of classes, each being specialized in reading a particular datale format. In this sense, classes derived from the base class datale can be developed to read NASTRAN input les, ANSYS les or any format corresponding to a traditional nite element program. It should be noted that, at this level, only the global functionality of the program is described, without mentioning the data structures which will sustain its implementation. This is typical for object oriented software, which is based on the interaction of objects rather than on the judicious choice of data structures.

3 3 ELEMENT AND NODE IMPLEMENTATION As mentioned before, in the proposed program structure, a separatation is made between the geometric description of the domain and the description of the differential equation, its boundary conditions and interpolation scheme. At the element and node level, this separation is evidenced by the existence of geometric and computable elements and geometric and computable nodes. The geometric element and node are responsible for the geometric description of a nite element. Its only responsibility towards the nite element computations is the generation of the jacobian matrix at a given integration point. This approach allows for the easy implementation of blending functions within the given program structure and for the dierential use of interpolation functions for geometric modeling and interpolation. Blending functions are particularly suitable for mapping analytically described curved boundary segments onto a square master region, a feature which is essential for the correct implementation of the p-adaptive method. The dierential use of families of shapefunctions for geometric modeling and interpolation is important because the geometric mapping best uses Lagrangian type shapefunctions whereas the better families for interpolation functions are hierarchical, such as Chebyshev or Legendre polynomials. ([6], [5]). The geometric node has an id, a coordinate system and the coordinates of the point associated with it. Its main function is to be used as a reference by the geometric elements. The computable element denes the element interpolation and implements the computation of the stiness matrix and boundary condition contribution. In its current implementation, the quadrilateral element implements arbitrary interpolation orders in the and directions of the master element. Corresponding to the interpolation orders the computable element contains two pointers to objects containing the Gauss integration points and weights. Each quadrilateral element has nine computable nodes, some of which may have zero degrees of freedom (e.g. in the case of linear interpolation). When an element is enriched in either of both directions, the number of degrees of freedom of the corresponding computable nodes is increased. The computable nodes have a unique id, an equation number and a vector of degrees of freedom. In the case of linear analysis, the values of the degrees of freedom are only used during post-processing. 4 ELEMENT INTEGRATION In almost all nite element programs, the element stiness is computed within an appropriate stiness routine. Each element implemented in the program has a separate stiness routine. Therefore, the completeness of a nite element program is often judged by the number of elements which are available. In the program presented, a dierent approach is taken, using the resources oered by the object oriented programming philosofy : a single stiness routine is written which is aplicable to any linear problem using a single integration rule. Such approach has the advantage that, once the p-adaptive algorithm has been implemented for one particular problem, it is available for all similar problems,

4 irrespective of the number of degrees of freedom per node. In a traditional nite element program, the material properties are passed on to the stiness routine as dierential equation coecients. These coecients are then used to integrate the stiness matrix and right hand side. In the approach presented, the stiness routine only performs the integration of a function returning matrix values. The function in this case is a method of the material class which, as a function of the shape function values and their derivatives, returns the corresponding matrix value. As the matrix class is an abstract concept independent of its size, the stiness routine presented is independent of the number of degrees of freedom per node and is therefore aplicable to any linear dierential equation. It is now irrelevant to ask how many elements are implemented. One should rather ask which material classes are available. Another advantage of this approach is that, whenever a new material is implemented, its approximations are automatically available in the p-adaptive context and, in the near future, the h-p-adaptive context as well. 5 BOUNDARY CONDITION IMPLEMENTA- TION In a traditional nite element program, the boundary conditions are invariably implemented on the global stiness matrix. Such approach is valid because all degrees of freedom have a physical meaning (e.g. displacement, temperature, pressure, etc.) and because the program performs only one type of simulation (heat transfer, elasticity, etc.). In p-adaptive methods, the shape functions are generally hierarchical and therefore the degrees of freedom lose their physical context. It is also impossible to apply Neumann or mixed boundary conditions at the level of the global stiness matrix. All boundary conditions (Dirichlet, Neumann and Mixed) are therefore implemented at the element level : the local stiness matrix and/or the right hand side are modied so that the global solution satises the given boundary conditions. The stiness matrix and right hand side are modied by integrating the proper coecients over the element sides. As in the stiness integration, the contributions to the stiness matrix are computed by the object of the material class, which, given the values of the shape functions and their derivatives on the boundary, computes the contribution to the element stiness matrix. This approach allows for the specication of boundary condition for elements with arbitrary number of shape functions and arbitrary sets of shape functions (e.g. Chebyshev, Legendre, etc.). It is also possible to apply this methodology for most problems in computational mechanics, linear or non-linear.

5 6 NODE NUMBERING, THE IMPLEMEN- TATION OF BLOCKED MATRICES As mentioned before, geometric nodes and computable nodes are implemented as separate classes. The computable nodes have a unique id, an equation number and a vector of degrees of freedom. Each computable node can have an arbitrary number of degrees of freedom, determined by the size of the vector of degrees of freedom. The notion of equation and equation number has been slightly extended in that an equation now refers to a group of algebraic equations corresponding to the degrees of freedom of a computable node. The concept of matrix was also extended to include blocked matrices. A blocked matrix can be viewed as rectangular matrix of submatrices. Each submatrix can have an arbitrary number of rows and columns, although it is assumed that the number of rows of the submatrices of a given row of the blocked matrix are identical and that the number of columns of the submatrices of a given column of a blocked matrix are also identical. The number of rows and/or columns of a submatrix can be zero, as is the case when the corresponding node has zero degrees of freedom. The data of the blocked matrix can be accessed in the traditional fashion as (i; j) denoting the element of the ith row and jth column, and also as (i; j; k; l) denoting the (k; l) element of submatrix (i; j). The blocked matrix includes a method to increase/decrease the number rows of the submatrices of a given row and/or increase/decrease the number columns of the submatrices of a given column. The underlying data structure of the blocked matrix is hidden from the user. In fact, two implementations were tested : a rst implementation in which the blocked matrix actually stores a matrix of objects of the class matrix and a second implementation in which the behaviour of the blocked matrix was implemented on top of the traditional storage scheme. The rst implementation is ecient for submatrix operations and is slower for indexing operations, whereas the second implementation is ecient for indexing and slower for submatrix operations. In the nite element program presented, the indexing operations clearly dominated and the second implementation was preferred. 7 THE IMPLEMENTATION OF THE P-RE- FINEMENT METHOD Given the structure of the program described in the previous sections, the p- renement algorithm is almost trivially implemented. As mentioned before, the quadrilateral computable elements have nine nodes, 4 nodes corresponding to the corners, 4 midside nodes and one node corresponding to the internal degrees of freedom. Each node can have an arbitrary number of degrees of freedom. The equations corresponding to a node are stored on a per node basis in a blocked matrix. When increasing/decreasing the polynomial order of either direction of the quadrilateral element, it is only needed to change the number of degrees of freedom of the corresponding nodes, without modifying the structure of the

6 blocked stiness matrix. Therefore, the equation (block) number of the computable nodes remains unchanged during a p-adaptive cycle, resulting in a very ecient adaptive code. This methodology is also trivially extended to triangles. The p-adaptive technique has been applied to several test cases described in [7] and yielded identical results as the examples mentioned in that work. No numerical results are presented here because the scope of this work is to describe the implementation of the p-adaptive element rather than to show numerical results. The real value of adaptive algorithms will only become aparent in the context of an interactive graphics environment. In a parallel eort, the rst author of this work is developing a graphics environment based on the object oriented programming philosofy with the perspective of incorporating the p-adaptive algorithm presented here. 8 CONCLUSIONS The object oriented programming philosofy enables the development of an innovative approach to nite element programming. The new structure presented is ideally suited for implementing the p-adaptive nite element method. It also allows for a single nite element program to be applied to virtually any linear simulation in the area of computational mechanics, irrespective of the number of degrees of freedom per node. It is believed that the data structure presented above takes advantage of the benets oered by the object oriented programming phylosofy and will form the basis of a powerful nite element program to be developed in the near future. References [1] Encarnac~ao J.L. and Lockemann, P.C. - Engeneering Databases. Springer Verlag, Berlin, [2] Forde, B. W. R., Foschi, R. O. and Stiemer, S. F. - Object Oriented Finite Element Analysis, Computers and Structures, 34, No 1, , (1990). [3] Alves Filho, J.S.R., Devloo, P.R.B. - Object Oriented Programming in Scientic Computing, The Beginning of a New Era, Engineering Computations, 8, (1991). [4] Devloo, P.R.B. and Alves Filho, J.S.R. - An Object Oriented Approach to Finite Element Programming (Phase 1) : A System Independent Windowing Environment for Developing Scientic Programs, Advances in Engineering Software and Workstations, to appear. [5] Devloo, P.R.B., Oden, J.T., Pattani, P. - An h-p Adaptive Finite Element Method for the Numerical Simulation of Compressible Flow, em Computer Methods in Applied Mechanics and Engineering, 70, , (1988). [6] Zienkiewicz, O.C., Craig, A. - Adaptive Renement, Error Estimates, Multigrid Solution, and Hierarchical Finite Element Method Concept, in Accuracy

7 Estimates and Adaptive Renements in Finite Element Computations, eds. Babuska, I., Zienkiewicz, O.C., Gago, J., Oliveira, E.R.de A., John Wiley & Sons, Chisester, [7] Duarte, C.A.M. Estudo da Vers~ao P do Metodo de Elementos Finitos para Problemas da Elasticidade e de Potencial, Master's Thesis, UFSC, Florianopolis, Brasil, November [8] Thomas, D. - What's in an object Byte, March, (1989). [9] Wiener, R.S., Pinson, L.J. An Introduction to Object Oriented Programming and C++. Addison Wesley, [10] Stroustrup, B. The C++ Programming Language. Addison Wesley, [11] Meyer, B. Object Oriented Software Construction. Prentice Hall, [12] CNS, Inc. C++/Views User Guide, software manual, 1991.

TICAM - Texas Institute for Computational and Applied Mathematics. The University of Texas at Austin. Taylor Hall Abstract

A New Cloud-Based hp Finite Element Method J. T. Oden, C. A. M. Duarte y and O. C. Zienkiewicz z TICAM - Texas Institute for Computational and Applied Mathematics The University of Texas at Austin Taylor