In this work we propose and analyze a novel Hybrid High-Order discretization of a class of (linear and) nonlinear elasticity models in the small deformation regime which are of common use in solid mechanics. The proposed method is valid in two and three space dimensions, it supports general meshes including polyhedral elements and nonmatching interfaces, enables arbitrary approximation order, and the resolution cost can be reduced by statically condensing a large subset of the unknowns for linearized versions of the problem. Additionally, the method satisfies a local principle of virtual work inside each mesh element, with interface tractions that obey the law of action and reaction. A complete analysis covering very general stress-strain laws is carried out, and optimal error estimates are proved. Extensive numerical validation on model test problems is also provided on two types of nonlinear models.
Robust and accurate schemes are designed to simulate the coupling between subsurface and overland flows. The coupling conditions at the interface enforce the continuity of both the normal flux and the pressure. Richards' equation governing the subsurface flow is discretized using a Backward Differentiation Formula and a symmetric interior penalty Discontinuous Galerkin method. The kinematic wave equation governing the overland flow is discretized using a Godunov scheme. Both schemes individually are mass conservative and can be used within single-step or multi-step coupling algorithms that ensure overall mass conservation owing to a specific design of the interface fluxes in the multi-step case. Numerical results are presented to illustrate the performances of the proposed algorithms.
The effects of uncertainty in hydrological laws are studied on subsurface flows modeled by Richards' equation. The empirical parameters of the water content and the hydraulic conductivity are considered as uncertain inputs of the model. One-dimensional infiltration problems are treated and the influence of the variability of the input parameters on the position and the spreading of the wetting front is evaluated. A Polynomial Chaos (PC) expansion is used to represent the output quantities and permits to significantly reduce the number of simulations in comparison with a classical Monte-Carlo method. A non-intrusive spectral projection supplies the coefficients of the PC decomposition. Three test cases with different hydrological laws are presented and demonstrate that second order PC expansions are sufficient to represent our quantities of interest owing to smooth dependences for the considered problems. Our results show a correlation between the position and the spreading of the wetting front and an amplification of the input uncertainty for all models. For each test case, five configurations with variable initial saturation state are investigated. The global sensitivity analysis indicates that the relative influence of an input parameter changes according to the output quantity considered and the initial saturation of the soil. The impact of the assumed distributions for the parameters is also briefly illustrated.
In this work we construct and analyze a nonconforming high-order discretization method for the quasi-static single-phase nonlinear poroelasticity problem describing Darcean flow in a deformable porous medium saturated by a slightly compressible fluid. The nonlinear elasticity operator is discretized using a Hybrid High-Order method, while the Darcy operator relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, supports arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle both nonzero and vanishing specific storage coefficients.
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