arXiv subjects

S. Schwarzacher

Publications and source records attributed to S. Schwarzacher.

8 recordsLinked to original sources

arXiv2024

Thermo-elastodynamics of nonlinearly viscous solids

In this paper, we study the thermo-elastodynamics of nonlinearly viscous solids in the Kelvin-Voigt rheology where both the elastic and the viscous stress tensors comply with the frame-indifference principle. The system features a force balance including inertia in the frame of nonsimple materials and a heat-transfer equation which is governed by the Fourier law in the deformed configuration. Combining a staggered minimizing movement scheme for quasi-static thermoviscoelasticity with a variational approach to hyperbolic PDEs, our main result consists in establishing the existence of weak solutions in the dynamic case. This is first achieved by including an additional higher-order regularization for the dissipation. Afterwards, this regularization can be removed by passing to a weaker formulation of the heat-transfer equation which complies with a total energy balance. The latter description hinges on regularity theory for the fourth order $p$-Laplacian which induces regularity estimates of the deformation beyond the standard estimates available from energy bounds. Besides being crucial for the proof, these extra regularity properties might be of independent interest and seem to be new in the setting of nonlinear viscoelasticity, also in the static or quasi-static case.

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arXiv2023

Geometric re-meshing strategies to simulate contactless rebounds of elastic solids in fluids

The paper deals with the rebound of an elastic solid off a rigid wall of a container filled with an incompressible Newtonian fluid. Our study focuses on a collision-free bounce, meaning a rebound without topological contact between the elastic solid and the wall. This has the advantage of omitting any artificial bouncing law. In order to capture the contact-free rebound for very small viscosities an adaptive numerical scheme is introduced. The here-introduced scheme is based on a Glowinski time scheme and a localized arbitrary Lagrangian-Eulerian map on finite elements in space. The absence of topological contact requires that very thin liquid channels are solved with sufficient accuracy. It is achieved via newly developed geometrically driven adaptive strategies. Using the numerical scheme, we present here a collection of numerical experiments. A rebound is simulated in the absence of topological contacts. Its physical relevance is demonstrated as, with decreasing viscosities, a free rebound in a vacuum is approached. Further, we compare the dynamics with a second numerical scheme; a here-introduced adaptive purely Eulerian level-set method. The scheme produced the same dynamics for large viscosities. However, as it requires a much higher computational cost, small viscosities can not be reached by this method. The experiments allow for a better understanding of the effect of fluids on the dynamics of elastic objects. Several observations are discussed, such as the amount of elastic and/or kinetic energy loss or the precise connection between the fluid pressure and the rebound of the solid.

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arXiv2022

Injectivity in second-gradient Nonlinear Elasticity

We study injectivity for models of Nonlinear Elasticity that involve the second gradient. We assume that $Ω\subset\mathbb{R}^n$ is a domain, $f\in W^{2,q}(Ω,\mathbb{R}^n)$ satisfies $|J_f|^{-a}\in L^1$ and that $f$ equals a given homeomorphism on $\partial Ω$. Under suitable conditions on $q$ and $a$ we show that $f$ must be a homeomorphism. As a main new tool we find an optimal condition for $a$ and $q$ that imply that $\mathcal{H}^{n-1}(\{J_f=0\})=0$ and hence $J_f$ cannot change sign. We further specify in dependence of $q$ and $a$ the maximal Hausdorff dimension $d$ of the critical set $\{J_f=0\}$. The sharpness of our conditions for $d$ is demonstrated by constructing respective counterexamples.

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arXiv2022

Variational methods for fluid-structure interaction and porous media

In this work we consider a poroelastic flexible material that may deform largely which is situated in an incompressible fluid driven by the Navier-Stokes equations in two or three space dimensions. By a variational approach we show existence of weak solutions for a class of such coupled systems. We consider the unsteady case, this means that the PDE for the poroelastic solid involves the Frechet derivative of a non-convex functional as well as (second order in time) inertia terms.

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arXiv2018

On global $L^q$ estimates for systems with $p$-growth in rough domains

We study regularity results for nonlinear parabolic systems of $p$-Laplacian type with inhomogeneous boundary and initial data, with $p\in(\frac{2n}{n+2},\infty)$. We show bounds on the gradient of solutions in the Lebesgue-spaces with arbitrary large integrability exponents and natural dependences on the right hand side and the boundary data. In particular, we provide a new proof of the global non-linear Calderón-Zygmund theory for such systems. Our method makes use of direct estimates on the solution minus its boundary values and hence is considerably shorter than the available higher integrability results. Technically interesting is the fact that our parabolic estimates have no scaling deficit with respect to the leading order term. Moreover, in the singular case, $p\in(\frac{2n}{n+2},2]$, any scaling deficit can be omitted.

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arXiv2016

Parabolic Lipschitz truncation and Caloric Approximation

We develop an improved version of the parabolic Lipschitz truncation, which allows qualitative control of the distributional time derivative and the preservation of zero boundary values. As a consequence, we establish a new caloric approximation lemma. We show functions. The distance is measured in terms of spatial gradients as well as almost uniformly in time. Both results are extended to the setting of Orlicz growth.

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arXiv2014

Finite element approximation of the $p(\cdot)$-Laplacian

We study a~priori estimates for the Dirichlet problem of the $p(\cdot)$-Laplacian, \[-\mathrm{div}(|\nabla v|^{p(\cdot)-2} \nabla v) = f. \] We show that the gradients of the finite element approximation with zero boundary data converges with rate $O(h^α)$ if the exponent $p$ is $α$-Hölder continuous. The error of the gradients is measured in the so-called quasi-norm, i.e. we measure the $L^2$-error of $|\nabla v|^{\frac{p-2}{2}} \nabla v$.

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arXiv2013

Solenoidal Lipschitz truncation for parabolic PDE's

We consider functions $u\in L^\infty(L^2)\cap L^p(W^{1,p})$ with $1<p<\infty$ on a time space domain. Solutions to non-linear evolutionary PDE's typically belong to these spaces. Many applications require a Lipschitz approximation $u_λ$ of $u$ which coincides with $u$ on a large set. For problems arising in fluid mechanics one needs to work with solenoidal (divergence-free) functions. Thus, we construct a Lipschitz approximation, which is also solenoidal. As an application we revise the existence proof for non-stationary generalized Newtonian fluids in [DRW10]. Since ${\rm div} u_λ=0$, we are able to work in the pressure free formulation, which heavily simplifies the proof. We also provide a simplified approach to the stationary solenoidal Lipschitz truncation of [BDF12].

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