Conforming Rectangular Mixed Finite Elements for Elasticity
TL;DR: A new family of rectangular mixed finite elements for the stress-displacement system of the plane elasticity problem is presented and it is proved that they are stable and error estimates for both the stress field and the displacement field are obtained.
Abstract: We present a new family of rectangular mixed finite elements for the stress-displacement system of the plane elasticity problem. Based on the theory of mixed finite element methods, we prove that they are stable and obtain error estimates for both the stress field and the displacement field. Using the finite element spaces in this family, an exact sequence is established as a discrete version of the elasticity complex in two dimensions. And the relationship between this discrete version and the original one is shown in a commuting diagram.
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TL;DR: In this article, a mixed finite element method for the two-dimensional Biot's consolidation model of poroelasticity is proposed, which uses the total stress tensor and fluid flux as primary unknown variables as well as the displacement and pore pressure.
Abstract: In this article, we propose a mixed finite element method for the two-dimensional Biot's consolidation model of poroelasticity. The new mixed formulation presented herein uses the total stress tensor and fluid flux as primary unknown variables as well as the displacement and pore pressure. This method is based on coupling two mixed finite element methods for each subproblem: the standard mixed finite element method for the flow subproblem and the Hellinger–Reissner formulation for the mechanical subproblem. Optimal a-priori error estimates are proved for both semidiscrete and fully discrete problems when the Raviart–Thomas space for the flow problem and the Arnold–Winther space for the elasticity problem are used. In particular, optimality in the stress, displacement, and pressure has been proved in when the constrained-specific storage coefficient is strictly positive and in the weaker norm when is nonnegative. We also present some of our numerical results.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1189–1210, 2014
86 citations
Cites methods from "Conforming Rectangular Mixed Finite..."
...In our simulations, we employ the rectangular Raviart–Thomas space of index 1 for the flow problem and the mixed finite element space developed by Chen and Wang [29] for the elasticity problem....
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TL;DR: In this article, a family of stable mixed finite elements for the linear elasticity on tetrahedral grids is constructed, where the stress is approximated by symmetric H(div)-P petertodd k−1 polynomial tensors and the displacement is estimated by C� −1-P�k€ p€ 1 polynomials, for all k ⩽ 4.
Abstract: A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed, where the stress is approximated by symmetric H(div)-P
k
polynomial tensors and the displacement is approximated by C
−1-P
k−1 polynomial vectors, for all k ⩽ 4. The main ingredients for the analysis are a new basis of the space of symmetric matrices, an intrinsic H(div) bubble function space on each element, and a new technique for establishing the discrete inf-sup condition. In particular, they enable us to prove that the divergence space of the H(div) bubble function space is identical to the orthogonal complement space of the rigid motion space with respect to the vector-valued P
k−1 polynomial space on each tetrahedron. The optimal error estimate is proved, verified by numerical examples.
70 citations
Posted Content•
TL;DR: In this paper, a family of mixed finite elements on triangular grids for solving the classical Hellinger-Reissner mixed problem of the elasticity equations is presented, and the wellposedness condition and the optimal a priori error estimate are proved for this family of finite elements.
Abstract: This paper presents a family of mixed finite elements on triangular grids for solving the classical Hellinger-Reissner mixed problem of the elasticity equations. In these elements, the matrix-valued stress field is approximated by the full $C^0$-$P_k$ space enriched by $(k-1)$ $H(\d)$ edge bubble functions on each internal edge, while the displacement field by the full discontinuous $P_{k-1}$ vector-valued space, for the polynomial degree $k\ge 3$. The main challenge is to find the correct stress finite element space matching the full $C^{-1}$-$P_{k-1}$ displacement space. The discrete stability analysis for the inf-sup condition does not rely on the usual Fortin operator, which is difficult to construct. It is done by characterizing the divergence of local stress space which covers the $P_{k-1}$ space of displacement orthogonal to the local rigid-motion. The well-posedness condition and the optimal a priori error estimate are proved for this family of finite elements. Numerical tests are presented to confirm the theoretical results.
56 citations
Cites background from "Conforming Rectangular Mixed Finite..."
...From then on, various stable mixed elements have been constructed, see [2, 4, 5, 9, 11, 17], [10, 19, 23, 27, 33, 34], and [8, 12, 18, 20, 21]....
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TL;DR: Arnabels et al. as discussed by the authors constructed lower order finite element subspaces of spaces of symmetric tensors with square-integrable divergence on a domain in any dimension.
Abstract: In this paper, we construct, in a unified fashion, lower order finite element subspaces of spaces of symmetric tensors with square-integrable divergence on a domain in any dimension. These subspaces are essentially the symmetric tensor finite element spaces of order k from [Finite element approximations of symmetric tensors on simplicial grids in ℝn: The higher order case, J. Comput. Math. 33 (2015) 283–296], enriched, for each (n − 1)-dimensional simplex, by (n+1)n 2 face bubble functions in the symmetric tensor finite element space of order n + 1 from [Finite element approximations of symmetric tensors on simplicial grids in ℝn: The higher order case, J. Comput. Math. 33 (2015) 283–296] when 1 ≤ k ≤ n − 1, and by (n−1)n 2 face bubble functions in the symmetric tensor finite element space of order n + 1 from [Finite element approximations of symmetric tensors on simplicial grids in ℝn: The higher order case, J. Comput. Math. 33 (2015) 283–296] when k = n. These spaces can be used to approximate the symmetric matrix field in a mixed formulation problem where the other variable is approximated by discontinuous piecewise Pk−1 polynomials. This in particular leads to first-order mixed elements on simplicial grids with total degrees of freedom per element 18 plus 3 in 2D, 48 plus 6 in 3D. The previous record of the degrees of freedom of first-order mixed elements is, 21 plus 3 in 2D, and 156 plus 6 in 3D, on simplicial grids. We also derive, in a unified way which is completely different from those used in [D. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions, Math. Comput. 77 (2008) 1229–1251; D. N. Arnold and R. Winther, Mixed finite element for elasticity, Number Math. 92 (2002) 401–419], a family of Arnold–Winther mixed finite elements in any space dimension. One example in this family is the Raviart–Thomas elements in one dimension, the second example is the mixed finite elements for linear elasticity in two dimensions due to Arnold and Winther, the third example is the mixed finite elements for linear elasticity in three dimensions due to Arnold, Awanou and Winther.
54 citations
TL;DR: A family of lower-order rectangular conforming mixed finite elements, in any space dimension, that shape function spaces for both stress and displacement are independent of the spatial dimension is constructed.
Abstract: We construct a family of lower-order rectangular conforming mixed finite elements, in any space dimension. In the method, the normal stress is approximated by quadratic polynomials $$\{1, x_{i}, x_{i}^{2}\}$$ { 1 , x i , x i 2 } , the shear stress by bilinear polynomials $$\{1, x_{i}, x_{j}, x_{i}x_{j}\}$$ { 1 , x i , x j , x i x j } , and the displacement by linear polynomials $$\{1, x_{i} \}$$ { 1 , x i } . The number of total degrees of freedom (dof) per element is 10 plus 4 in 2D, and 21 plus 6 in 3D, while the previous record of least dof for conforming element is 17 plus 4 in 2D, and 72 plus 12 in 3D. The feature of this family of elements is, besides simplicity, that shape function spaces for both stress and displacement are independent of the spatial dimension $$n$$ n . As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. The well-posedness condition and the optimal a priori error estimate are proved. Numerical tests show the stability and effectiveness of these new elements.
50 citations
Cites background or methods from "Conforming Rectangular Mixed Finite..."
...Based on such a condition, conforming mixed finite elements on the simplical and rectangular triangulations are developed for both 2D and 3D [1,3,4,8,10,16]....
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...For the rectangular partition, the 2D simplest conforming element in the literature has 17 stress and 4 displacement dof per element [16], and in 3D the simplest conforming element has 72 plus 12 dof per element [10]....
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References
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Book•
01 Jan 1978TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Abstract: Preface 1. Elliptic boundary value problems 2. Introduction to the finite element method 3. Conforming finite element methods for second-order problems 4. Other finite element methods for second-order problems 5. Application of the finite element method to some nonlinear problems 6. Finite element methods for the plate problem 7. A mixed finite element method 8. Finite element methods for shells Epilogue Bibliography Glossary of symbols Index.
8,407 citations
Book•
01 Apr 2002
TL;DR: In this article, Ciarlet presents a self-contained book on finite element methods for analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces.
Abstract: From the Publisher:
This book is particularly useful to graduate students, researchers, and engineers using finite element methods. The reader should have knowledge of analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces. Other than these basics, the book is mathematically self-contained.
About the Author
Philippe G. Ciarlet is a Professor at the Laboratoire d'Analyse Numerique at the Universite Pierre et Marie Curie in Paris. He is also a member of the French Academy of Sciences. He is the author of more than a dozen books on a variety of topics and is a frequent invited lecturer at meetings and universities throughout the world. Professor Ciarlet has served approximately 75 visiting professorships since 1973, and he is a member of the editorial boards of more than 20 journals.
8,052 citations
"Conforming Rectangular Mixed Finite..." refers background in this paper
...Remark 2 Q0 K is the Bogner-Fox-Schmit element [12]....
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Book•
23 Nov 2011TL;DR: Variational Formulations and Finite Element Methods for Elliptic Problems, Incompressible Materials and Flow Problems, and Other Applications.
Abstract: Variational Formulations and Finite Element Methods. Approximation of Saddle Point Problems. Function Spaces and Finite Element Approximations. Various Examples. Complements on Mixed Methods for Elliptic Problems. Incompressible Materials and Flow Problems. Other Applications.
5,030 citations
"Conforming Rectangular Mixed Finite..." refers methods in this paper
...Based on the theory of mixed finite element methods [9, 10], we get ‖σ − σh‖H(div,Ω) + ‖u − uh‖0,Ω ≤ c ( inf τh∈Σh ‖σ − τh‖H(div,Ω) + inf vh∈Vh ‖u − vh‖0,Ω ) ....
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...From the theory of mixed methods [9, 10], the following two stability conditions ensure the existence and uniqueness of the solution to this discrete problem, as well as a good approximation to the exact solution....
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TL;DR: In this article, the authors present two families of non-conforming finite elements, built on tetrahedrons or on cubes, which are respectively conforming in the spacesH(curl) and H(div).
Abstract: We present here some new families of non conforming finite elements in ?3. These two families of finite elements, built on tetrahedrons or on cubes are respectively conforming in the spacesH(curl) andH(div). We give some applications of these elements for the approximation of Maxwell's equations and equations of elasticity.
3,049 citations
"Conforming Rectangular Mixed Finite..." refers methods in this paper
...The space (V l K) has dimension 2(l + 1)(l + 2)− 3 by using the fact that the kernel of is the space of infinitesimal rigid motions [17]....
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