Hybrid High-Order methods for nonlinear solid mechanics

Abstract : In this thesis, we are interested in the devising of Hybrid High-Order (HHO) methods for nonlinear solid mechanics. HHO methods are formulated in terms of face unknowns on the mesh skeleton. Cell unknowns are also introduced for the stability and approximation properties of the method. HHO methods offer several advantages in solid mechanics: \textit{(i)} primal formulation; \textit{(ii)} free of volumetric locking due to incompressibility constraints; \textit{(iii)} arbitrary approximation order $k\geq1$ ; \textit{(iv)} support of polyhedral meshes with possibly non-matching interfaces; and \textit{(v)} attractive computational costs due to the static condensation to eliminate locally cell unknowns while keeping a compact stencil. In this thesis, primal HHO methods are devised to solve the problem of finite hyperelastic deformations and small plastic deformations. An extension to finite elastoplastic deformations is also presented within a logarithmic strain framework. Finally, a combination with Nitsche's approach allows us to impose weakly the unilateral contact and Tresca friction conditions. Optimal convergence rates of order $h^{k+1}$ are proved in the energy-norm. All these methods have been implemented in both the open-source library DiSk++ and the open-source industrial software code_aster. Various two- and three-dimensional benchmarks are considered to validate these methods and compare them with $H^1$-conforming and mixed finite element methods.
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Nicolas Pignet. Hybrid High-Order methods for nonlinear solid mechanics. Numerical Analysis [math.NA]. Université Paris-Est Marne la Vallée, 2019. English. ⟨tel-02318157⟩

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