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Mixed hp finite element methods for problems in elasticity and Stokes flow
, 1994
"... We consider the mixed formulation for the elasticity problem and the limiting Stokes problem in IR d , d = 2; 3. We derive a set of sufficient conditions under which families of mixed finite element spaces are simultaneously stable with respect to the mesh size h and, subject to a maximum loss of ..."
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Cited by 16 (4 self)
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We consider the mixed formulation for the elasticity problem and the limiting Stokes problem in IR d , d = 2; 3. We derive a set of sufficient conditions under which families of mixed finite element spaces are simultaneously stable with respect to the mesh size h and, subject to a maximum loss of O(k d\Gamma1 2 ), with respect to the polynomial degree k. We obtain asymptotic rates of convergence that are optimal up to O(k ffl ) in the displacement/velocity and up to O(k d\Gamma1 2 +ffl ) in the "pressure", with ffl ? 0 arbitrary (both rates being optimal with respect to h). Several choices of elements are discussed with reference to properties desirable in the context of the hpversion. Key Words: hp, mixed method, incompressible elasticity, Stokes problem. AMS(MOS) subject classifications (1985 revision): 65N30 Permanent address: Faculty of Mechanical Engineering, Helsinki University of Technology, 02150 Esbo, Finland. email: stenberg@hut.fi. y Department of Mathematics...
Analytic and Computational Assessment of Locking in the hp Finite Element Method
 Comput. Methods Appl. Mech. Engrg
, 1994
"... Locking is the phenomenon by which the numerical approximation of parameterdependent problems deteriorates for values of the parameter close to a limiting value. In this paper, we give a definition of locking and develop precise computable and analytic ways to quantify it. Using the example of near ..."
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Cited by 8 (3 self)
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Locking is the phenomenon by which the numerical approximation of parameterdependent problems deteriorates for values of the parameter close to a limiting value. In this paper, we give a definition of locking and develop precise computable and analytic ways to quantify it. Using the example of nearly incompressible elasticity, we show by means of computational and theoretical results, the difference between the h version and p=hp version in combatting locking. Our results establish the superiority of high order elements (both h; p and hp) when the standard variational form is used. We also discuss other issues such as curved elements, mixed methods, and locking phenomena for problems over anisotropic materials and over thin domains. Key Words: locking, h version, p version, hp version, finite element method To appear in Computer Methods in Applied Mechanics and Engineering, 1995. (1) Research partially supported by the Air Force Office of Scientific Research, Bolling AFB, DC, under G...
FirstOrder System Least Squares for Linear Elasticity: Numerical Results
"... . The aim here is to study two firstorder system leastsquares (FOSLS) methods applied to various boundary value problems of planar linear elasticity. Both use finite element discretization and multigrid solution methods. They are twostage algorithms that first solve for the displacement flux vari ..."
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Cited by 3 (3 self)
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. The aim here is to study two firstorder system leastsquares (FOSLS) methods applied to various boundary value problems of planar linear elasticity. Both use finite element discretization and multigrid solution methods. They are twostage algorithms that first solve for the displacement flux variable (the gradient of displacement, which easily yields the deformation and stress variables), then for displacement variable itself. As a complement to a companion theoretical paper, this paper focuses on numerical results, including finite element accuracy and multigrid convergence estimates that confirm uniform optimal performanceeven as the material tends to the incompressible limit. Key words. Elasticity equations, firstorder system least squares, Lam'e constants, multigrid AMS subject classifications. 65F10, 65N55, 73V05 1. Introduction. Let\Omega be a convex polygon or a C 1;1 domain in R 2 with boundary \Gamma. Denote the Lam'e constants by ¯ and , where (¯; ) 2 [¯ 1 ; ¯ ...
An Adaptive Finite Element Technique With aPriori Mesh Grading
, 1993
"... . In this paper we discuss a combined apriori aposteriori approach to mesh refinement in finite element methods for two and threedimensional elliptic boundary value problems containing boundary singularities. We review first both techniques of apriori mesh grading around singularities and apos ..."
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Cited by 3 (1 self)
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. In this paper we discuss a combined apriori aposteriori approach to mesh refinement in finite element methods for two and threedimensional elliptic boundary value problems containing boundary singularities. We review first both techniques of apriori mesh grading around singularities and aposteriori mesh refinement controlled by local error indicators. In examples of two and threedimensional boundary value problems we demonstrate the applicability and efficiency of various combined mesh refinement strategies. Key Words. Elliptic boundary value problem, finite element method, adaptive mesh refinement, aposteriori error estimation, apriori mesh grading, singularities. Supported by DAAD (German Academic Exchange Service), No. 517/009/511/3. y Supported by DAAD (German Academic Exchange Service), No. 517/009/503/3. 1 Introduction The quality of a finite element approximation to the solution of an elliptic boundary value problem can vary markedly over the computational do...
Multigrid Methods for the Pure Traction Problem of Linear Elasticity: Mixed Formulation
 SIAM Journal on Numerical Analysis
, 1998
"... For the twodimensional pure traction boundary value problem of linear elasticity, a multigrid method using conforming P 1 finite element is developed. Moreover, it is used to solve the nonconforming discretization of the pure traction problem as a coarsergrid correction in multigrid algorithm. In ..."
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Cited by 2 (0 self)
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For the twodimensional pure traction boundary value problem of linear elasticity, a multigrid method using conforming P 1 finite element is developed. Moreover, it is used to solve the nonconforming discretization of the pure traction problem as a coarsergrid correction in multigrid algorithm. In both cases the convergence is uniform even as the material becomes nearly incompressible. A heuristic argument for acceleration of the multigrid method is discussed and computational results are included. 1 Introduction Let\Omega be a bounded convex polygonal domain in R 2 and @\Omega = S n i=1 \Gamma i . The pure traction boundary value problem for planar linear elasticity is given in the form \Gamma div ¸ n 2¯ ffl ß (u ¸ ) + tr i ffl ß (u ¸ ) j ffi ß o = f ¸ in\Omega ; (1) i 2¯ ffl ß (u ¸ ) + tr i ffl ß (u ¸ ) j ffi ß j ¸ i j \Gamma i = g ¸ i ; 1 i n ; (2) where u ¸ denotes the displacement, f ¸ the body force, g ¸ i the boundary tracti...
The Current Status of Unsteady CFD Approaches for Aerodynamic Flow Control
, 2002
"... An overview of the current status of time dependent algorithms is presented. Special attention is given to algorithms used to predict #uid actuator #ows, as well as other active and passive#ow control devices. Capabilities for the next decade are predicted, and principal impediments to the progress ..."
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Cited by 1 (0 self)
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An overview of the current status of time dependent algorithms is presented. Special attention is given to algorithms used to predict #uid actuator #ows, as well as other active and passive#ow control devices. Capabilities for the next decade are predicted, and principal impediments to the progress of timedependent algorithms are identi#ed.
Crack Growth Simulation and Residual Strength Prediction in Airplane Fuselages
, 1999
"... this report is derived primarily from the Ph.D. thesis of the #rst author #19#. ..."
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this report is derived primarily from the Ph.D. thesis of the #rst author #19#.
A Variationally Consistent Mesh Adaptation Method for Triangular Elements in Explicit Lagrangian Dynamics
"... In this paper a variational formulation for mesh adaptation procedures, involving local mesh changes for triangular meshes, is presented. Such local adaptive changes are very well suited for explicit methods as they do not involve significant computational expense. They also greatly simplify the pro ..."
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In this paper a variational formulation for mesh adaptation procedures, involving local mesh changes for triangular meshes, is presented. Such local adaptive changes are very well suited for explicit methods as they do not involve significant computational expense. They also greatly simplify the projection of field variables from the old to the new meshes. Crucially, the variational nature of the formulation used to derive the equilibrium equations at steps where adaptation takes place ensures that conservation of linear and angular momentum is obtained [1]. Several examples in 2D showing the application of the proposed adaptive algorithms are used to demonstrate the validity of the methodology proposed. Copyright c ○ 2008 John Wiley & Sons, Ltd.