542 research outputs found
Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft
Computational Engineering
The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods
A locking-free discontinuous Galerkin method for linear elastic Steklov eigenvalue problem
In this paper, a discontinuous Galerkin finite element method of Nitsche's
version for the Steklov eigenvalue problem in linear elasticity is presented.
The a priori error estimates are analyzed under a low regularity condition, and
the robustness with respect to nearly incompressible materials (locking-free)
is proven. Furthermore, some numerical experiments are reported to show the
effectiveness and robustness of the proposed method.Comment: 25 pages, 6 figure
A phase-field model for fractures in incompressible solids
Within this work, we develop a phase-field description for simulating
fractures in incompressible materials. Standard formulations are subject to
volume-locking when the solid is (nearly) incompressible. We propose an
approach that builds on a mixed form of the displacement equation with two
unknowns: a displacement field and a hydro-static pressure variable.
Corresponding function spaces have to be chosen properly. On the discrete
level, stable Taylor-Hood elements are employed for the displacement-pressure
system. Two additional variables describe the phase-field solution and the
crack irreversibility constraint. Therefore, the final system contains four
variables: displacements, pressure, phase-field, and a Lagrange multiplier. The
resulting discrete system is nonlinear and solved monolithically with a
Newton-type method. Our proposed model is demonstrated by means of several
numerical studies based on two numerical tests. First, different finite element
choices are compared in order to investigate the influence of higher-order
elements in the proposed settings. Further, numerical results including spatial
mesh refinement studies and variations in Poisson's ratio approaching the
incompressible limit, are presented
A cell-based smoothed finite element method for kinematic limit analysis
This paper presents a new numerical procedure for kinematic limit analysis problems, which incorporates the cell-based smoothed finite element method with second-order cone programming. The application of a strain smoothing technique to the standard displacement finite element both rules out volumetric locking and also results in an efficient method that can provide accurate solutions with minimal computational effort. The non-smooth optimization problem is formulated as a problem of minimizing a sum of Euclidean norms, ensuring that the resulting optimization problem can be solved by an efficient second-order cone programming algorithm. Plane stress and plane strain problems governed by the von Mises criterion are considered, but extensions to problems with other yield criteria having a similar conic quadratic form or 3D problems can be envisaged
Recovered finite element methods
We introduce a family of Galerkin finite element methods which are constructed via recovery operators over element-wise discontinuous approximation spaces. This new family, termed collectively as \emph{recovered finite element methods (R-FEM)} has a number of attractive features over both classical finite element and discontinuous Galerkin approaches, most important of which is its potential to produce stable conforming approximations in a variety of settings. Moreover, for special choices of recovery operators, R-FEM produces the same approximate solution as the classical conforming finite element method, while, trivially, one can recast (primal formulation) discontinuous Galerkin methods. A priori error bounds are shown for linear second order boundary value problems, verifying the optimality of the proposed method. Residual-type a posteriori bounds are also derived, highlighting the potential of R-FEM in the context of adaptive computations. Numerical experiments highlight the good approximation properties of the method in practice. A discussion on the potential use of R-FEM in various settings is also included
A Locking-Free Weak Galerkin Finite Element Method for Linear Elasticity Problems
In this paper, we introduce and analyze a lowest-order locking-free weak
Galerkin (WG) finite element scheme for the grad-div formulation of linear
elasticity problems. The scheme uses linear functions in the interior of mesh
elements and constants on edges (2D) or faces (3D), respectively, to
approximate the displacement. An -conforming displacement
reconstruction operator is employed to modify test functions in the right-hand
side of the discrete form, in order to eliminate the dependence of the
parameter in error estimates, i.e., making the scheme
locking-free. The method works without requiring to be bounded. We prove optimal error estimates, independent of
, in both the -norm and the -norm. Numerical experiments
validate that the method is effective and locking-free
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