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Register a new userA Hybrid High-Order method for creeping flows of non-Newtonian fluids
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Added by
Andr\\'e Harnist
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this paper, we design and analyze a Hybrid High-Order discretization method for the steady motion of non-Newtonian, incompressible fluids in the Stokes approximation of small velocities. The proposed method has several appealing features including the support of general meshes and high-order, unconditional inf-sup stability, and orders of convergence that match those obtained for Leray-Lions scalar problems. A complete well-posedness and convergence analysis of the method is carried out under new, general assumptions on the strain rate-shear stress law, which encompass several common examples such as the power-law and Carreau-Yasuda models. Numerical examples complete the exposition.
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We propose a finite element method for simulating one-dimensional solid models moving and experiencing large deformations while immersed in generalized Newtonian fluids. The method is oriented towards applications involving microscopic devices or organisms in the soft-bio-matter realm. By considering that the strain energy of the solid may explicitly depend on time, we incorporate a mechanism for active response. The solids are modeled as Cosserat rods, a detailed formulation being provided for the special case of a planar non-shearable rod. The discretization adopts one-dimensional Hermite elements for the rod and low-order Lagrange two-dimensional elements for the fluids velocity and pressure. The fluid mesh is boundary-fitted, with remeshing at each time step. Several time marching schemes are studied, of which a semi-implicit scheme emerges as most effective. The method is demonstrated in very challenging examples: the roll-up of a rod to circular shape and later sudden release, the interaction of a soft rod with a fluid jet and the active self-locomotion of a sperm-like rod. The article includes a detailed description of a code that implements the method in the Firedrake library.
We present a residual-based a posteriori error estimator for the hybrid high-order (HHO) method for the Stokes model problem. Both the proposed HHO method and error estimator are valid in two and three dimensions and support arbitrary approximation orders on fairly general meshes. The upper bound and lower bound of the error estimator are proved, in which proof, the key ingredient is a novel stabilizer employed in the discrete scheme. By using the given estimator, adaptive algorithm of HHO method is designed to solve model problem. Finally, the expected theoretical results are numerically demonstrated on a variety of meshes for model problem.
We propose an Extended Hybrid High-Order scheme for the Poisson problem with solution possessing weak singularities. Some general assumptions are stated on the nature of this singularity and the remaining part of the solution. The method is formulated by enriching the local polynomial spaces with appropriate singular functions. Via a detailed error analysis, the method is shown to converge optimally in both discrete and continuous energy norms. Some tests are conducted in two dimensions for singularities arising from irregular geometries in the domain. The numerical simulations illustrate the established error estimates, and show the method to be a significant improvement over a standard Hybrid High-Order method.
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 lineariz
We design a Hybrid High-Order (HHO) scheme for the Poisson problem that is fully robust on polytopal meshes in the presence of small edges/faces. We state general assumptions on the stabilisation terms involved in the scheme, under which optimal error estimates (in discrete and continuous energy norms, as well as $L^2$-norm) are established with multiplicative constants that do not depend on the maximum number of faces in each element, or the relative size between an element and its faces. We illustrate the error estimates through numerical simulations in 2D and 3D on meshes designed by agglomeration techniques (such meshes naturally have elements with a very large numbers of faces, and very small faces).
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