TY - JOUR
T1 - A hybridizable discontinuous Galerkin formulation for non-linear elasticity
AU - Kabaria, Hardik
AU - Lew, Adrian J.
AU - Cockburn, Bernardo
N1 - Publisher Copyright:
© 2014 Elsevier B.V.
PY - 2015/1/1
Y1 - 2015/1/1
N2 - We revisit the hybridizable discontinuous Galerkin method for non-linear elasticity introduced by S.-C. Soon (2008). We show that it can be recast as a minimization problem of a non-linear functional over a space of discontinuous approximations to the displacement. The functional can be written as the sum over the elements of the classic potential energy plus a new energy associated to the inter-element jumps of the displacement. We then show that if this new energy is not properly weighted, the minimizers might not converge to the exact solution. We construct an example illustrating this phenomenon and show how to overcome it by suitably increasing the weight of the energy of the inter-element jumps. Finally, we explore the performance of the method for the case of piecewise-linear approximations in rather demanding situations in both two-dimensional and, for the first time, three-dimensional situations. They include almost incompressible materials, large deformations with large-shear layers, and cavitation. We also compare the method with the continuous Galerkin method and a previously explored discontinuous Galerkin method, and show that, when using piecewise-linear approximations and a moderate number of degrees of freedom, the current method turns out to be more efficient for the computation of the gradient.
AB - We revisit the hybridizable discontinuous Galerkin method for non-linear elasticity introduced by S.-C. Soon (2008). We show that it can be recast as a minimization problem of a non-linear functional over a space of discontinuous approximations to the displacement. The functional can be written as the sum over the elements of the classic potential energy plus a new energy associated to the inter-element jumps of the displacement. We then show that if this new energy is not properly weighted, the minimizers might not converge to the exact solution. We construct an example illustrating this phenomenon and show how to overcome it by suitably increasing the weight of the energy of the inter-element jumps. Finally, we explore the performance of the method for the case of piecewise-linear approximations in rather demanding situations in both two-dimensional and, for the first time, three-dimensional situations. They include almost incompressible materials, large deformations with large-shear layers, and cavitation. We also compare the method with the continuous Galerkin method and a previously explored discontinuous Galerkin method, and show that, when using piecewise-linear approximations and a moderate number of degrees of freedom, the current method turns out to be more efficient for the computation of the gradient.
KW - Cavitation
KW - Hybridized discontinuous Galerkin
KW - Large shear deformations
KW - Non-convergence
KW - Non-linear elasticity
KW - Stabilization
UR - http://www.scopus.com/inward/record.url?scp=84908388667&partnerID=8YFLogxK
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U2 - 10.1016/j.cma.2014.08.012
DO - 10.1016/j.cma.2014.08.012
M3 - Article
AN - SCOPUS:84908388667
SN - 0045-7825
VL - 283
SP - 303
EP - 329
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
ER -