Computer Methods in Applied Mechanics and Engineering, 2009
... Received 6 May 2008; revised 30 September 2008; accepted 31 December 2008. Available online 8... more ... Received 6 May 2008; revised 30 September 2008; accepted 31 December 2008. Available online 8 January 2009. Abstract. This paper presents a methodology for generating high-order shape functions for the complete family of finite elements. ...
Descrevem-se de m,aneira formal os conceitos matemáticos que permitem implementar um refinamento ... more Descrevem-se de m,aneira formal os conceitos matemáticos que permitem implementar um refinamento multidimensional h-p adaptativo....
Definem-se o padrão de refinamento de elementos em 1, 2 e 3 dimensões. Elementos como um interval... more Definem-se o padrão de refinamento de elementos em 1, 2 e 3 dimensões. Elementos como um intervalo. triângulos, quadriláteros, tetraedros, hexaedros, pirâmides e prismas. Aliado a essa subdivisão geométrica se desenvolve adaptatividade H e P para elementos trabalhando em uma malha hibrida com elementos de múltiplas dimensões. Se mostra como gerar desta forma um espaço de interpolação de funções pelo menos contínuas. Os elementos aqui definidos foram implementados em um programa de elementos finitos multidisciplinar onde diversos problemas foram resolvidos. O programa é de distribuição livre e está disponível na página de FEC Faculdade de Engenharia Civil, Arquitetura e Urbanismo da UNICAMP Universidade Estadual de Campinas. O programa é denominado PZ e foi desenvolvido na Filosofia de Programação Orientada Objetos C++, e também roda em paralelo. Dois artigos sobre este trabalho foram publicados se destacando definição de todas as funções de forma para grau de interpolação p qualquer...
This contribution presents an overview of current research being developed at the laboratory for ... more This contribution presents an overview of current research being developed at the laboratory for computational mechanics (LABMEC) at the Faculdade de Engenharia Civil - UNICAMP - Brasil. A tridimensional hp-adaptive environment permits the user to create hp-adaptive interpolation spaces for use in nite element simulations. The hp-adaptive elements can be rened in a preestablished pattern, or the user can instruct the program of which pattern to use. In another line of research on adaptive methods, the capability of superposing small scale meshes on top of large scale meshes should allow to model multiscale problems. Finally, the main results on the development of fully automated procedures to adapt n-dimensional (n<=3) meshes are presented.
Resumo. hp refinement applied to the finite element method should lead to faster convergence than... more Resumo. hp refinement applied to the finite element method should lead to faster convergence than h or p refinement [10] and [11]. On the other hand, h and p refinements are easier and faster to implement. The complexity of an hp refinement is increased when the process is implemented for any type of mesh [3] In the PZ environment [4], the hp adaptive method was designed to work with heterogenous mesh including 1D, 2D and 3D elements. For industrial purposes, the adaptive method has a performance problem, because the re-finement process, the error estimation and the choice of the refinement pattern (i.e. given a certain number of degree of freedom choose how to increase h, p or hp) have a high computa-tional cost. In this work, an implementation of the mesh refinement using elements based on templates will be discussed. Template based implementation presents faster codes than derivate classes / virtual calls. The performance problem has several points to focus on, the main point bei...
This contribution presents an overview of current research being developed at the laboratory for ... more This contribution presents an overview of current research being developed at the laboratory for computational mechanics (LABMEC) at the Faculdade de Engenharia Civil - UNICAMP - Brasil. A tridimensional hp- adaptive environment permits the user to create hp-adaptive interpolation spaces for use in nite element simulations. The hp-adaptive elements can be rened in a preestablished pattern, or the user can
ABSTRACT This article shows in detail how to construct in a simple and ordered way a set of ratio... more ABSTRACT 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.
This contribution presents an overview of current research being developed at the laboratory for ... more This contribution presents an overview of current research being developed at the laboratory for computational mechanics (LABMEC) at the Faculdade de Engenharia Civil - UNICAMP - Brasil. A tridimensional hp- adaptive environment permits the user to create hp-adaptive interpolation spaces for use in nite element simulations. The hp-adaptive elements can be rened in a preestablished pattern, or the user can instruct the program of which pattern to use. In another line of research on adaptive methods, the capability of superposing small scale meshes on top of large scale meshes should allow to model multiscale problems. Finally, the main results on the development of fully automated procedures to adapt n-dimensional (n 3) meshes are presented.
This paper is devoted to the development of a model for the numerical simulation of hydraulic fra... more 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.
Computer Methods in Applied Mechanics and Engineering, 2009
... Received 6 May 2008; revised 30 September 2008; accepted 31 December 2008. Available online 8... more ... Received 6 May 2008; revised 30 September 2008; accepted 31 December 2008. Available online 8 January 2009. Abstract. This paper presents a methodology for generating high-order shape functions for the complete family of finite elements. ...
Computer Methods in Applied Mechanics and Engineering, 2009
... Received 6 May 2008; revised 30 September 2008; accepted 31 December 2008. Available online 8... more ... Received 6 May 2008; revised 30 September 2008; accepted 31 December 2008. Available online 8 January 2009. Abstract. This paper presents a methodology for generating high-order shape functions for the complete family of finite elements. ...
Descrevem-se de m,aneira formal os conceitos matemáticos que permitem implementar um refinamento ... more Descrevem-se de m,aneira formal os conceitos matemáticos que permitem implementar um refinamento multidimensional h-p adaptativo....
Definem-se o padrão de refinamento de elementos em 1, 2 e 3 dimensões. Elementos como um interval... more Definem-se o padrão de refinamento de elementos em 1, 2 e 3 dimensões. Elementos como um intervalo. triângulos, quadriláteros, tetraedros, hexaedros, pirâmides e prismas. Aliado a essa subdivisão geométrica se desenvolve adaptatividade H e P para elementos trabalhando em uma malha hibrida com elementos de múltiplas dimensões. Se mostra como gerar desta forma um espaço de interpolação de funções pelo menos contínuas. Os elementos aqui definidos foram implementados em um programa de elementos finitos multidisciplinar onde diversos problemas foram resolvidos. O programa é de distribuição livre e está disponível na página de FEC Faculdade de Engenharia Civil, Arquitetura e Urbanismo da UNICAMP Universidade Estadual de Campinas. O programa é denominado PZ e foi desenvolvido na Filosofia de Programação Orientada Objetos C++, e também roda em paralelo. Dois artigos sobre este trabalho foram publicados se destacando definição de todas as funções de forma para grau de interpolação p qualquer...
This contribution presents an overview of current research being developed at the laboratory for ... more This contribution presents an overview of current research being developed at the laboratory for computational mechanics (LABMEC) at the Faculdade de Engenharia Civil - UNICAMP - Brasil. A tridimensional hp-adaptive environment permits the user to create hp-adaptive interpolation spaces for use in nite element simulations. The hp-adaptive elements can be rened in a preestablished pattern, or the user can instruct the program of which pattern to use. In another line of research on adaptive methods, the capability of superposing small scale meshes on top of large scale meshes should allow to model multiscale problems. Finally, the main results on the development of fully automated procedures to adapt n-dimensional (n<=3) meshes are presented.
Resumo. hp refinement applied to the finite element method should lead to faster convergence than... more Resumo. hp refinement applied to the finite element method should lead to faster convergence than h or p refinement [10] and [11]. On the other hand, h and p refinements are easier and faster to implement. The complexity of an hp refinement is increased when the process is implemented for any type of mesh [3] In the PZ environment [4], the hp adaptive method was designed to work with heterogenous mesh including 1D, 2D and 3D elements. For industrial purposes, the adaptive method has a performance problem, because the re-finement process, the error estimation and the choice of the refinement pattern (i.e. given a certain number of degree of freedom choose how to increase h, p or hp) have a high computa-tional cost. In this work, an implementation of the mesh refinement using elements based on templates will be discussed. Template based implementation presents faster codes than derivate classes / virtual calls. The performance problem has several points to focus on, the main point bei...
This contribution presents an overview of current research being developed at the laboratory for ... more This contribution presents an overview of current research being developed at the laboratory for computational mechanics (LABMEC) at the Faculdade de Engenharia Civil - UNICAMP - Brasil. A tridimensional hp- adaptive environment permits the user to create hp-adaptive interpolation spaces for use in nite element simulations. The hp-adaptive elements can be rened in a preestablished pattern, or the user can
ABSTRACT This article shows in detail how to construct in a simple and ordered way a set of ratio... more ABSTRACT 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.
This contribution presents an overview of current research being developed at the laboratory for ... more This contribution presents an overview of current research being developed at the laboratory for computational mechanics (LABMEC) at the Faculdade de Engenharia Civil - UNICAMP - Brasil. A tridimensional hp- adaptive environment permits the user to create hp-adaptive interpolation spaces for use in nite element simulations. The hp-adaptive elements can be rened in a preestablished pattern, or the user can instruct the program of which pattern to use. In another line of research on adaptive methods, the capability of superposing small scale meshes on top of large scale meshes should allow to model multiscale problems. Finally, the main results on the development of fully automated procedures to adapt n-dimensional (n 3) meshes are presented.
This paper is devoted to the development of a model for the numerical simulation of hydraulic fra... more 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.
Computer Methods in Applied Mechanics and Engineering, 2009
... Received 6 May 2008; revised 30 September 2008; accepted 31 December 2008. Available online 8... more ... Received 6 May 2008; revised 30 September 2008; accepted 31 December 2008. Available online 8 January 2009. Abstract. This paper presents a methodology for generating high-order shape functions for the complete family of finite elements. ...
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