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Register a new userA mass conserving mixed stress formulation for Stokes flow with weakly imposed stress symmetry
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We introduce a new discretization of a mixed formulation of the incompressible Stokes equations that includes symmetric viscous stresses. The method is built upon a mass conserving mixed formulation that we recently studied. The improvement in this work is a new method that directly approximates the viscous fluid stress $sigma$, enforcing its symmetry weakly. The finite element space in which the stress is approximated consists of matrix-valued functions having continuous normal-tangential components across element interfaces. Stability is achieved by adding certain matrix bubbles that were introduced earlier in the literature on finite elements for linear elasticity. Like the earlier work, the new method here approximates the fluid velocity $u$ using $H(operatorname{div})$-conforming finite elements, thus providing exact mass conservation. Our error analysis shows optimal convergence rates for the pressure and the stress variables. An additional post processing yields an optimally convergent velocity satisfying exact mass conservation. The method is also pressure robust.
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We propose a new discretization of a mixed stress formulation of the Stokes equations. The velocity $u$ is approximated with $H(operatorname{div})$-conforming finite elements providing exact mass conservation. While many standard methods use $H^1$-conforming spaces for the discrete velocity, $H(operatorname{div})$-conformity fits the considered variational formulation in this work. A new stress-like variable $sigma$ equalling the gradient of the velocity is set within a new function space $H(operatorname{curl} operatorname{div})$. New matrix-valued finite elements having continuous normal-tangential components are constructed to approximate functions in $H(operatorname{curl} operatorname{div})$. An error analysis concludes with optimal rates of convergence for errors in $u$ (measured in a discrete $H^1$-norm), errors in $sigma$ (measured in $L^2$) and the pressure $p$ (also measured in $L^2$). The exact mass conservation property is directly related to another structure-preservation property called pressure robustness, as shown by pressure-independent velocity error estimates. The computational cost measured in terms of interface degrees of freedom is comparable to old and new Stokes discretizations.
A stress equilibration procedure for linear elasticity is proposed and analyzed in this paper with emphasis on the behavior for (nearly) incompressible materials. Based on the displacement-pressure approximation computed with a stable finite element pair, it constructs an $H (text{div})$-conforming, weakly symmetric stress reconstruction. Our focus is on the Taylor-Hood combination of continuous finite element spaces of polynomial degrees $k+1$ and $k$ for the displacement and the pressure, respectively. Our construction leads then to reconstructed stresses by Raviart-Thomas elements of degree $k$ which are weakly symmetric in the sense that its anti-symmetric part is zero tested against continuous piecewise polynomial functions of degree $k$. The computation is performed locally on a set of vertex patches covering the computational domain in the spirit of equilibration cite{BraSch:08}. Due to the weak symmetry constraint, the local problems need to satisfy consistency conditions associated with all rigid body modes, in contrast to the case of Poissons equation where only the constant modes are involved. The resulting error estimator is shown to constitute a guaranteed upper bound for the error with a constant that depends only on the shape regularity of the triangulation. Local efficiency, uniformly in the incompressible limit, is deduced from the upper bound by the residual error estimator.
A posteriori error estimates are constructed for the three-field variational formulation of the Biot problem involving the displacements, the total pressure and the fluid pressure. The discretization under focus is the H1({Omega})-conforming Taylor-Hood finite element combination, consisting of polynomial degrees k + 1 for the displacements and the fluid pressure and k for the total pressure. An a posteriori error estimator is derived on the basis of H(div)-conforming reconstructions of the stress and flux approximations. The symmetry of the reconstructed stress is allowed to be satisfied only weakly. The reconstructions can be performed locally on a set of vertex patches and lead to a guaranteed upper bound for the error with a constant that depends only on local constants associated with the patches and thus on the shape regularity of the triangulation. Particular emphasis is given to nearly incompressible materials and the error estimates hold uniformly in the incompressible limit. Numerical results on the L-shaped domain confirm the theory and the suitable use of the error estimator in adaptive strategies.
Hydrodynamics of the non-relativistic compressible fluid in the curved spacetime is derived using the generalized framework of the stochastic variational method (SVM) for continuum medium. The fluid-stress tensor of the resultant equation becomes asymmetric for the exchange of the indices, differently from the standard Euclidean one. Its incompressible limit suggests that the viscous term should be represented with the Bochner Laplacian. Moreover the modified Navier-Stokes-Fourier (NSF) equation proposed by Brenner can be considered even in the curved spacetime. To confirm the compatibility with the symmetry principle, SVM is applied to the gauge-invariant Lagrangian of a charged compressible fluid and then the Lorentz force is reproduced as the interaction between the Abelian gauge fields and the viscous charged fluid.
The enrichment formulation of double-interpolation finite element method (DFEM) is developed in this paper. DFEM is first proposed by Zheng emph{et al} (2011) and it requires two stages of interpolation to construct the trial function. The first stage of interpolation is the same as the standard finite element interpolation. Then the interpolation is reproduced by an additional procedure using the nodal values and nodal gradients which are derived from the first stage as interpolants. The re-constructed trial functions are now able to produce continuous nodal gradients, smooth nodal stress without post-processing and higher order basis without increasing the total degrees of freedom. Several benchmark numerical examples are performed to investigate accuracy and efficiency of DFEM and enriched DFEM. When compared with standard FEM, super-convergence rate and better accuracy are obtained by DFEM. For the numerical simulation of crack propagation, better accuracy is obtained in the evaluation of displacement norm, energy norm and the stress intensity factor.
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