1,720,961 research outputs found
Elastodynamics
No full textThe goal of this chapter is to show how the HHO method can be used for the space semi-discretization of the elastic wave equation. For simplicity, we restrict the scope to media undergoing infinitesimal deformations and governed by a linear stress-strain constitutive relation. We consider first the second-order formulation in time and then the mixed formulation leading to a first-order formulation in time. The time discretization is realized, respectively, by means of Newmark schemes and diagonally-implicit or explicit Runge–Kutta schemes. Interestingly, considering the mixed-order HHO method is instrumental to devise explicit Runge–Kutta schemes
Linear Elasticity and Hyperelasticity
No full textIn this chapter, we show how to discretize using HHO methods linear elasticity and nonlinear hyperelasticity problems. In particular, we pay particular attention to the robustness of the discretization in the quasi-incompressible limit. For linear elasticity, we reconstruct the strain tensor in the space composed of symmetric gradients of vector-valued polynomials. For nonlinear hyperelasticity, we reconstruct the deformation gradient in a full tensor-valued polynomial space, and not just in a space composed of polynomial gradients. We also consider a second gradient reconstruction in an even larger space built using Raviart–Thomas polynomials, for which no additional stabilization is necessary. Finally, we present some numerical examples
Plasticity
No full textModeling plasticity problems is particularly relevant in nonlinear solid mechanics since plasticity can have a major influence on the behavior of a mechanical structure. One difficulty is that the plastic deformations are generally assumed to be incompressible, leading to volume-locking problems if (low-order) H1 -conforming finite elements are used. Mixed methods avoid these problems, but need additional globally coupled unknowns to enforce the incompressibility of the plastic deformations
Mathematical Aspects
No full textThe objective of this chapter is to put the HHO method presented in the previous chapter on a firm mathematical ground. In particular, we prove the key stability and convergence results announced in the previous chapter
Getting Started: Linear Diffusion
No full textThe objective of this chapter is to gently introduce the hybrid high-order (HHO) method on one of the simplest model problems: the Poisson problem with homogeneous Dirichlet boundary conditions. Our goal is to present the key ideas underlying the devising of the method and state its main properties (most of them without proof). The keywords of this chapter are cell and face unknowns, local reconstruction and stabilization operators, elementwise assembly, static condensation, energy minimization, and equilibrated fluxes
Some Variants
No full textThe goal of this chapter is to explore some variants of the HHO method devised in Chap. 1 and analyzed in Chap. 2. We first study two variants of the gradient reconstruction operator that will turn useful, for instance, when dealing with nonlinear problems in Chaps. 4 and 7. Then, we explore a mixed-order variant of the HHO method that is useful, for instance, to treat domains with a curved boundary. Finally, we bridge the HHO method to the finite element and virtual element viewpoints
Contact and Friction
No full textIn this chapter, we show how the HHO method can be used to discretize a linear elasticity problem with nonlinear boundary conditions resulting from contact and friction. The main idea is to use a boundary penalty technique to enforce these conditions. This approach leads, under some assumptions, to a discrete semilinear form enjoying a monotonicity property. The error analysis reveals that the degree of the face unknowns on the contact/friction boundary has to be raised to (k+ 1 ) to ensure optimal estimates
Implementation Aspects
No full textIn this chapter, we outline the steps needed to bring the abstract formulation of the HHO method to an actual implementation. For simplicity, we focus on the Poisson model problem (see Chap. 1 ). We show how the local HHO operators (reconstruction and stabilization) are translated into matrices that can be used in the actual computation, and we give some criteria to test the implementation. Then we discuss the assembly of the discrete problem and the handling of the boundary conditions. We conclude with a brief overview on computational costs. Along the chapter, we provide some snippets of Matlab®/Octave code to show a possible implementation (in 1D) of the critical parts
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