Extended Bond Graph Representation of the Traction Problem in Linear Elastodynamics

1991 ◽  
Vol 113 (1) ◽  
pp. 118-121 ◽  
Author(s):  
M. E. Ingrim ◽  
G. Y. Masada
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Solid Mechanics ◽  
A Priori ◽  
Bond Graph ◽  
Elasticity Tensor ◽  
Graph Representation ◽  
Approximation Techniques ◽  
Elastic Bodies ◽  
Linear Elastodynamics

To illustrate the use of the extended bond graph notation, a reticulation is developed for a conjugate variable approximation of the traction problem in linear elastodynamics. This reticulation is general in the sense that all vector and tensor quantities are expressed using direct notation; that is, no specific coordinate system is chosen a priori. In addition, the only limitation placed upon the elasticity tensor C(X) is that it be symmetric. This allows homogeneous and inhomogeneous isotropic, orthotropic, etc., linearly elastic bodies to be modeled using these results. The conjugate approximations used here are entirely compatible with Galerkin based finite element methods. Consequently, this extended bond graph reticulation allows well-developed approximation techniques in solid mechanics to be directly incorporated into bond graph based system models.

scholarly journals Convergence Analysis of H(div)-Conforming Finite Element Methods for a Nonlinear Poroelasticity Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Yuping Zeng ◽  
Zhifeng Weng ◽  
Fen Liang
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
A Priori ◽  
Mixed Finite Elements ◽  
Elastic Displacement ◽  
Fully Discrete ◽  
Conforming Finite Element ◽  
Priori Error Estimates ◽  
Galerkin Formulation ◽  
Flow Variables

In this paper, we introduce and analyze H(div)-conforming finite element methods for a nonlinear model in poroelasticity. More precisely, the flow variables are discretized by H(div)-conforming mixed finite elements, while the elastic displacement is approximated by the H(div)-conforming finite element with the interior penalty discontinuous Galerkin formulation. Optimal a priori error estimates are derived for both semidiscrete and fully discrete schemes.

scholarly journals Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions

2018 ◽  
Vol 52 (6) ◽  
pp. 2247-2282 ◽  
Author(s):  
Erik Burman ◽  
Peter Hansbo ◽  
Mats G. Larson ◽  
André Massing
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Stiffness Matrix ◽  
A Priori ◽  
Beltrami Operator ◽  
Optimal Order ◽  
Codimension One ◽  
Special Cases ◽  
Arbitrary Codimension ◽  
Background Mesh

We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in ℝd of arbitrary codimension. The method is based on using continuous piecewise linears on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in ℝ3.

scholarly journals Recovered finite element methods on polygonal and polyhedral meshes

2020 ◽  
Vol 54 (4) ◽  
pp. 1309-1337
Author(s):  
Zhaonan Dong ◽  
Emmanuil H. Georgoulis ◽  
Tristan Pryer
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Degrees Of Freedom ◽  
A Priori ◽  
Diffusion Reaction ◽  
Galerkin Methods ◽  
Polyhedral Meshes ◽  
Order Polynomial ◽  
Spatial Dimensions ◽  
Practical Performance

Recovered Finite Element Methods (R-FEM) have been recently introduced in Georgoulis and Pryer [Comput. Methods Appl. Mech. Eng. 332 (2018) 303–324]. for meshes consisting of simplicial and/or box-type elements. Here, utilising the flexibility of the R-FEM framework, we extend their definition to polygonal and polyhedral meshes in two and three spatial dimensions, respectively. An attractive feature of this framework is its ability to produce arbitrary order polynomial conforming discretizations, yet involving only as many degrees of freedom as discontinuous Galerkin methods over general polygonal/polyhedral meshes with potentially many faces per element. A priori error bounds are shown for general linear, possibly degenerate, second order advection-diffusion-reaction boundary value problems. A series of numerical experiments highlight the good practical performance of the proposed numerical framework.

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