Modeling Using The Finite Element Method And Its Object Oriented Implementation

The philosophy of approximation of the finite element method and its corresponding object oriented implementation is discussed. It is shown, based on examples, how the finite element method can be used to approximate most of the conservation laws. Next, it is shown how this philosophy of approximation of the finite element method is translated in an object oriented programming environment suitable for the development of finite element algorithms. Finally, recent developments of the author's research group will be presented, including a discussion of auto-adaptive techniques applied to finite element approximations.1736Bangerth, W., Using modern features of C++ for adaptive finite element methods: Dimension-independent programming in deal.II (2000) Proceedings of the 16th MACS World Congress 2000, Lausanne, Switzerland, 2000, , Michel Deville and Robert Owens, editors, Document Sessions/118-1Becker, E.B., Tinsley Oden, J., Carey, G.F., (1983) Finite Elements: An Introduction, 1. , Prentice-HallBravo, C.M.A.A., (2000) Sobre a Implementação da Técnica Hp-adaptativa Tri-dimensional Para Elementos Finitos, , PhD thesis, Faculdade de Engenharia Mecânica, UNICAMPBravo, C.M.A.A., Devloo, P.R.B., Pavanello, R., Sobre a implementação do refinamento h-p (1999) CILAMCE99 Congresso Ibero Latino Americana de Metodos Computacionais Para Engenharia, pp. 1-21. , Sao PauloDevloo, P.R.B., Oden, J.T., Pattani, P., An hp-adaptive finite element method for the numerical simulation of compressible flow (1988) Computer Methods in Applied Mechanics and Engineering, 70, pp. 203-235Devloo, P.R.B., Object oriented tools for scientific computing (2000) Engineering with Computers, 16, pp. 63-72Devloo, P.R.B., Bravo, C.M.A., An object oriented approach to adaptive finite element techniques (2000) European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS, pp. 1-21Devloo, P.R.B., Longhin, G.C., An object oriented multi-threaded multi-frontal solver (2001) XXII Iberian Latin-American Congress on Computational Methods in EngineeringDubois, P.F., (1997) Object Technology for Scientific ComputingRank, E., Krause, R., A multi-scale finite-element method (1997) Computers & Structures, (64), pp. 139-144Gurtin, M.E., (1981) An Introduction to Continuum Mechanics, , Academic PressHaney, S.W., Beating the abstraction penalty in C++ using expression templates (1996) Computers in Physics, 10 (6), pp. 552-557. , Nov/DecVeldhuizen, T.L., Rapid linear algebra in C++ (1996) Dr. Dobb's Journal, , Augus

An Object Oriented Frameword For Flexible Mechanism Simulation

An object oriented framework is described which allows to model flexible mechanisms, simply by composing the flexible components and their links as assembled elements. The issue of the parameter space is also addressed 1.Cardona, A., Geradin, M., Modelling of superelements in mechanism analysis (1991) International Journal on Numerical Methods in Engineering, 32, pp. 1565-1593Chung, J., Hulbert, G.M., A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized/alpha method (1993) ASME J. Appl. Mech., 60, pp. 371-375Devloo, P.R.B., Géradin, M., Fleury, R., A Corotational Formulation for the Simulation of Flexible Mechanisms. Multibody System Dynamics, , page Accepted for publicationGeradin, M., Rixen, D., (1994) Mechanica Vibrations Theory and Applications to Structural Dynamics, , WileyHughes, T.J.R., (1987) The Finite Element Method : Linear Static and Dynamic Finite Element Analysis, , Prentice Hall, Englewood Cliffs, New JerseyHulbert, G.M., Jang, I., Automatic time step control algorithms for structural dynamics (1995) CMAME, 126, pp. 155-17

On An Object Oriented Approach To Adaptive Finite Element Techniques

This paper describes a object oriented implementation of the h-p-adaptive finite element method for one- two- and three-dimensional approximations. The adaptive algorithm is implemented within the PZ environment and includes linear, triangular, quadrilateral, cube, tetrahedral, prismatic and pyramidal elements. This large number of elements can be refined within a single program because of the object oriented nature of the approach. Rather than concentrating on the data structure which allows to implement the different elements and refinements, the behaviour necessary to implement the refinement is emphasized.Oden, J.T., Demkowicz, L.F., (1996) Applied Functional Analysis, , CRC Press, Boca Raton, New York, London, TokyoDemkowicz, L., Devloo, P.R.B., Oden, J.T., On an h-type mesh refinement strategy based on minimization of interpolation errors (1986) Computer Methods in Applied Mechanics and Engineering, 1-2, pp. 63-87Devloo, P.R.B., Pz : An object oriented environment for scientific programming (1997) Computer Methods in Applied Mechanics and Engineering, 150, pp. 133-153Devloo, P.R.B., A three-dimensional adaptive finite element strategy (1991) Computers & Structures, 38 (2), pp. 121-130Devloo, P.R.B., (1987) An H-P Adaptive Finite Element Method for Steady Compressible Flow, , PhD thesis, The University of Texas at Austin, AugustRheinboldt, W.C., Babuska, I., Error estimates for adaptive finite element computations (1978) Siam J. Numer. Anal., 15, pp. 736-754. , augustRachowicz, W., Westermann, T.A., Oden, J.T., Demkowics, L., Toward a universal hp adaptive finite element strategy - part 2, Constrained approximation and data structure (1989) Comput. Methods Appl. Mech. Engrg., 77, pp. 113-180Kreyszig, E., (1978) Introductory Functional Analysis with Applications, , John Wiley & SonsRachowicz, W., Hardy, O., Demkowics, L., Oden, J.T., Toward a universal hp adaptive finite element strategy - part1. Constrained approximation and data structure (1989) Comput. Methods Appl. Mech. Engrg., 77, pp. 79-112Sherma, A.H., Bank, R.E., An adaptive, multi-level metho for elliptic boundary value problems (1981) Computing, 26, pp. 91-105Babuska, I., Suri, M., The p and h-p versions of the finite elements approximations: Analysis of the optimal mesh sequences in one dimension (1990) Comput. Methods Appl. Mech. Engrg., 80, pp. 5-26Zienkiewics, O.C., Zhu, J.Z., A simple error estimator and adaptive procedure for a practical engineering analysis (1987) Int. J. Num. Methods Engrg., 24, pp. 337-35

Wavelets And Adaptive Grids For The Discontinuous Galerkin Method

In this paper, space adaptivity is introduced to control the error in the numerical solution of hyperbolic systems of conservation laws. The reference numerical scheme is a new version of the discontinuous Galerkin method, which uses an implicit diffusive term in the direction of the streamlines, for stability purposes. The decision whether to refine or to unrefine the grid in a certain location is taken according to the magnitude of wavelet coefficients, which are indicators of local smoothness of the numerical solution. Numerical solutions of the nonlinear Euler equations illustrate the efficiency of the method. © Springer 2005.391-3143154Abgrall, R., Harten, A., Multiresolution representation in unstructured meshes (1998) SIAM J. Numer. Anal.Bihari, B.L., Harten, A., Multiresolution schemes for the numerical solution of 2-D conservation laws I (1997) SIAM J. Sci. Comput., 18 (2)Bonhaus, D.L., (1998) A Higher Order Accurate Finite Element Method for Viscous Compressible Flows, , Ph.D. thesis, Virginia Polytechnics Institute and State University (November)Brooks, A., Hughes, T., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations (1982) Comput. Methods Appl. Mech. Engrg., 32Chiavassa, G., Donat, R., Numerical experiments with multilevel schemes for conservation laws (1999) Godunov's Methods: Theory and Applications, , ed. Toro (Kluwer Academic/Plenum, Dordrecht)Cockburn, B., Shu, C.-W., Runge-Kutta discontinuous Galerkin method for convection-dominated problems (2001) J. Sci. Comput., 16Cohen, A., Muller, S., Postel, M., Ould-Kabe, S.M., Fully adaptive multiresolution finite volume schemes for conservation laws (2002) Math. Comp., 72Dahmen, W., Gottschlich-Müller, B., Müller, S., Multiresolution schemes for conservation laws (1998) Numer. Math., 88Díaz Calle, J.L., Devloo, P.R.B., Gomes, S.M., Stabilized discontinuous Galerkin method for hyperbolic equations Comput. Methods Appl. Mech. Engrg., , to appearDomingues, M.O., Gomes, S.M., Diaz, L.A., Adaptive wavelet representation and differentiation on block-structured grids (2003) Appl. Numer. Math., 8 (3-4)Harten, A., Adaptive multiresolution schemes for shock computations (1994) J. Comput. Phys., 115Harten, A., Multiresolution representation of data: A general framework (1996) SIAM J. Numer. Anal., 33Holmström, M., (1997) Wavelet Based Methods for Time Dependent PDE, , Ph.D. thesis, Uppsala University, SwedenKaibara, M.K., Gomes, S.M., Fully adaptive multiresolution scheme for shock computations (1999) Godunov's Methods: Theory and Applications, , ed. Toro (Kluwer Academic/Plenum, Dordrecht)Sjögreen, B., Numerical experiments with the multiresolution schemes for the compressible Euler equations (1995) J. Comput. Phys., 117Vasilyev, O.V., Bowman, C., Second generation wavelet collocation method for the solution of partial differential equations (2000) J. Comput. Phys., 165Waldén, J., Filter bank methods for hyperbolic PDEs (1999) SIAM J. Numer. Anal., 3

Uma tecnica elemento por elemento livre de matrics para a avaliaçao de resíduos

Neste trabalho propoe-se um novo melhoramento técnica elemento por elemento (EPE), que se utiliza da deficiencia de posto natural das matrizes de elemento. Dessa forma, o cálculo dos residuos necessário na solucio iterativa do sistema de equaçoes pode ser efetuado sem a avaliaçáo explícita das matrizes de elemento. Os produtos matriz por vetor siio otimizados para minimizar o número de operaçóes de ponto flutuante e a área de memória. A técnica resultante constitui-se em um método livre de matrizes no sentido em que niio há necessidade de armazenamento para as matrizes de elemento, o que prove um método poderoso para a abordagem de problemas de grandes dimensoes. O desempenho da técnica EPE Livre de Matrizes é avaliado na soluciio de problemas simétricos e nao simétricos em diversos ambientes computacionais, desde estacoes de trabalho até supercomputadores paralelos

Enriched two dimensional mixed finite element models for linear elasticity with weak stress symmetry

The purpose of this article is to derive and analyze new discrete mixed approximations for linear elasticity problems with weak stress symmetry. These approximations are based on the application of enriched versions of classic Poisson-compatible spaces, for stress and displacement variables, and/or on enriched Stokes-compatible space configurations, for the choice of rotation spaces used to weakly enforce stress symmetry. Accordingly, the stress space has to be adapted to ensure stability. Such enrichment procedures are done via space increments with extra bubble functions, which have their support on a single element (in the case of H1-conforming approximations) or with vanishing normal components over element edges (in the case of H(div)-conforming spaces). The advantage of using bubbles as stabilization corrections relies on the fact that all extra degrees of freedom can be condensed, in a way that the number of equations to be solved and the matrix structure are not affected. Enhanced approximations are observed when using the resulting enriched space configurations, which may have different orders of accuracy for the different variables. A general error analysis is derived in order to identify the contribution of each kind of bubble increment on the accuracy of the variables, individually. The use of enriched Poisson spaces improves the rates of convergence of stress divergence and displacement variables. Stokes enhancement by bubbles contributes to equilibrate the accuracy of weak stress symmetry enforcement with the stress approximation order, reaching the maximum rate given by the normal traces (which are not affected)7992678270

A Finite Element Model For Three Dimensional Hydraulic Fracturing

This paper is devoted to the development of a model for the numerical simulation of hydraulic fracturing processes with 3d fracture propagation. It takes into account the efects of fluid flow inside the fracture, fluid leak-off through fracture walls and elastic response of the surrounding porous media. Finite element techniques are adopted for the discretization of the conservation law of fluid flow and the singular integral equation relating the traction and the fracture opening. The discrete model for the singular integral equation is implemented using a stencil matrix structure allowing an efficient implementation of the fluid-structure interaction problem. Newton's method combined with GMRES linear solver are used to solve the resulting nonlinear set of equations. An algorithm for fracture propagation is proposed which is based on the balance of the amount of fluid transported to a certain point with the amount of fluid that could be lost by leak-off. To illustrate the feasibility of the model, we present simulation results for typical operational parameters. © 2006 IMACS.731-4 SPEC. ISS.142155Bui, H.D., An integral equation method for solving the problem of a plane crack of arbitrary shape (1977) J. Mech. Phys. Solids, 25, pp. 29-39Carey, G.F., Yew, C.H., Ouyang, S., An adaptive moving boundary technique for fluid-fracture propagation (1997) Int. J. Num. Methods Fluids, 24, pp. 645-670Carter, R.D., Optimum fluid characteristics for fracture extension (1957) ASME Spring Meeting, Mid-Continent District, , ASME, Tulas, OKGu, H., Yew, C.H., Finite element solution of a boundary integral equation for mode i embedded three dimensional fractures (1988) Int. J. Numer. Methods Eng., 26, pp. 1525-1540Perkins, T.K., Kern, L.R., Widths of hydraulic fractures (1961) J. Petrol. Technol., pp. 937-94

Hierarchical High Order Finite Element Bases For H (div) Spaces Based On Curved Meshes For Two-dimensional Regions Or Manifolds

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)The mixed finite element formulation for elliptic problems is characterized by simultaneous calculations of the potential (primal variable) and of the flux field (dual variable). This work focuses on new H(div)-conforming finite element spaces, which are suitable for flux approximations, based on curved meshes of a planar region or a manifold domain embedded in R3. The adopted methodology for the construction of H(div) bases consists in using hierarchical H1-conforming scalar bases multiplied by vector fields that are properly constructed on the master element and mapped to the geometrical elements by the Piola transformation, followed by a normalization procedure. They are classified as being of edge or internal type. The normal component of an edge function coincides on the corresponding edge with the associated scalar shape function, and vanishes over the other edges, and the normal components of an internal shape function vanishes on all element edges. These properties are fundamental for the global assembly of H(div)-conforming functions locally defined by these vectorial shape functions. For applications to the mixed formulation, the configuration of the approximation spaces is such that the divergence of the dual space and the primal approximation space coincides. Results of verification numerical tests are presented for curved triangular and quadrilateral partitions on circular, cylindrical and spherical regions, demonstrating stable convergence with optimal convergence rates, coinciding for primal and dual variables. © 2016 Elsevier B.V. All rigths reserved.3012412582013/21959-4, FAPESP, Conselho Nacional de Desenvolvimento Científico e Tecnológico308632/2006-0, CNPq, Conselho Nacional de Desenvolvimento Científico e Tecnológico310369/2006-1, CNPq, Conselho Nacional de Desenvolvimento Científico e TecnológicoFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq

Definition Of A P-interpolating Space Of Hierarchical Bases Of Finite Elements On The Pyramid

This article shows in detail how to construct in a simple and ordered way a set of rational functions defined on the pyramid topology. The set of functions is parameterized by an integer p. It is shown that these functions, defined in a hierarchical way, constitute a basis for a complete polynomial interpolation space of degree p on the pyramid domain. In order to help this definition we use a denumerable sequence of orthogonal polynomials defined on an interval of the real line. A priori, any increasing sequence of polynomials in one variable can be used. The bases are constructed to be used in the class C0 method of finite elements. The rational functions thus defined can be combined to represent any polynomial of degree p. Thus, given an arbitrary number p, one defines a finite element whose geometry is a pyramid that has associated a complete interpolation space of degree p. Moreover, this element is adequate to be used with the p-adaptive technique on heterogeneous meshes of finite elements hierarchical. © 2014 Elsevier Inc.460174204Ayala Bravo, C.M.A., (2000) Um Sistema de Refinamento H-p Adaptativo Utilizando Elementos Finitos Hierárquicos Multidimensionais, , Departamento de Mecânica Computacional da Faculdade de Engenharia Mecânica, UNICAMP Campinas, BrasilBabuska, I., Szabo, B.A., Katz, I.N., The p-version of the finite element method (1981) SIAM J. Numer. Anal., 18 (3), pp. 515-545Campion, S.D., Jarvis, J.L., An investigation of the implementation of the p-version finite element method (1996) Finite Elem. Anal. Des., 23, pp. 1-21Choquet, G., Topology (1966) Pure Appl. Math., 19, p. 503. , ISBN-13: 978-0121734503, Copyright 2010 Elsevier Ltd. All rights reservedDevloo, P.R.B., Ayala Bravo, C.M.A., Rylo, E.C., Systematic and generic construction of shape functions for p-adaptive meshes of multidimensional finite elements (2009) Comput. Methods Appl. Mech. Engrg., 5, pp. 1716-1725Hoffman, R.M., Busby, H.R., Stress-based convergence analysis for p-adaptive hierarchical finite element analysis (2000) Finite elements in analysis and design, 34 (1), pp. 25-35. , DOI 10.1016/S0168-874X(99)00025-6Lax, P.D., Linear Algebra (1997) Pure Appl. Math., , John Wiley & Sons, Inc. 0-471-11111-2Liu, Y., Busby, H.R., P-version hybrid/mixed finite element method (1998) Finite Elements in Analysis and Design, 30 (4), pp. 325-333. , PII S0168874X98000420Owen, S.J., Saigal, S., Formation of pyramid elements for hexahedra to tetrahedra transitions (2001) Computer Methods in Applied Mechanics and Engineering, 190 (34), pp. 4505-4518. , DOI 10.1016/S0045-7825(00)00330-3, PII S0045782500003303Sherwin, S.J., Hierarchical hp finite elements in hybrid domains (1997) Finite Elem. Anal. Des., 27, pp. 109-119Vijayakar, S.M., Busby, H.R., Houser, D.R., Finite element analysis of quasi-prismatic bodies using Chebyshev polynomials (1987) Internat. J. Numer. Methods Engrg., 24, pp. 1461-147

An Improved Numerical Integration Algorithm For Elastoplastic Constitutive Equations

A simplified methodology is proposed for elastoplastic calculations which holds for associative models. It is based on the representation of the elastoplastic model based on a rotation of the principal stresses and the fact that, in such system of coordinates, the direction that minimizes the square of a distance between a trial stress and the plastic surface has the same direction as the plastic deformation evolution. Such an approach allows for the elastoplastic calculation of complex models to be simpler and more efficient computationally. The proposed methodology is verified by the application to the elastoplastic model of Sandler-DiMaggio.6419Souza Neto, E., Peric, D., Owen, D.R.J., (2008) Computational Methods for Plasticity, , John Wiley & Sons LtdChen, W.F., Han, D.J., (1988) Plasticity for Structural Engineers, , Springer-VerlagDiMaggio, F.L., Sandler, I., Material model for granular soils (1971) Eng Mech Div, 97 (3), pp. 935-950Devloo, P.R.B., PZ: an object oriented environment for scientific programming (1997) Comput Methods Appl Mech Eng, 150 (1-4), pp. 133-153Devloo, P.R.B., Object oriented tools for scientific computing (2000) Eng Comput, 46 (2), pp. 203-214Lainé, E., Vallée, C., Fortuné, D., Nonlinear isotropic constitutive laws: choice of the three invariants, convex potentials and constitutive inequalities (1999) Int J Eng Sci, 37 (15), pp. 1927-1941Borja, R.I., (2012) Plasticity Modeling & Computation, , Springer Science & Business MediaArmero, F., Pérez-Foguet, A., On the formulation of closest-point projection algorithms in elastoplasticity - Part I: The variational structure (2002) Int J Numer Methods Eng, 53 (2), pp. 297-329Armero, F., Pérez-Foguet, A., On the formulation of closest-point projection algorithms in elastoplasticity - Part II: Globally convergent schemes (2002) Int J Numer Methods Eng, 53 (2), pp. 331-374Foster, C.D., Regueiro, R.A., Fossum, A.F., Borja, R.I., Implicit numerical integration of a three-invariant, isotropic/kinematic hardening cap plasticity model for geomaterials (2005) Comput Methods Appl Mech Eng, 194, pp. 5109-5138Fossum, A.F., Fredrich, J.T., Cap plasticity models and compactive and dilatant pre-failure deformation (2000), 4th North American Rock Mechanics Symposium, Seatle, United States of AmericaFossum, A.F., Brannon, R.M., The Sandia Geomodel: Theory and Users Guide (2004), SAND Report, Sandia National LaboratoriesSandler, I.S., Rubin, D., An algorithm and a modular subroutine for the cap model (1979) Int J Numer Anal Methods Geomech, 3 (2), pp. 173-18