Relaxation methods and finite element schemes for the equations of visco-elastodynamics. Chiara Simeoni

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1 Relaxation methods and finite element schemes for the equations of visco-elastodynamics Chiara Simeoni Department of Information Engineering, Computer Science and Mathematics University of L Aquila (Italy) Polytech Nice Sophia - University of Nice Sophia Antipolis (France) discussion in progress with C. Lattanzio and A.E. Tzavaras preliminary issues of finite element relaxation schemes for elastodynamics from physical modeling through analytical studies for numerical purposes relaxation approximations support entropy inequalities dissipative mechanism and damping effects on oscillations stability/convergence of finite elements schemes computational performance adaptive meshing yields an extra stabilization against the nonlinear response over shock regions

2 diffusive stress relaxation for elastodynamics in one space dimension deformation gradient u ε (t, x) R and velocity v ε (t, x) R kinematic viscosity µ = µ(ε)>0, relaxation parameter ε>0 non-homogeneous (strictly) hyperbolic 3 3 semilinear system ut ε vx ε = 0 vt ε Zx ε = 0 εzt ε µ vx ε = Z ε + S(u ε ) with the corresponding system for the original fields (one-field equation) { u ε t v ε x = 0 v ε t S(u ε ) x µ v ε xx + ε v ε tt = 0 from the 2 2 incompletely parabolic system of visco-elastodynamics supplementary relaxation variable Z ε (t, x) R for the Piola-Kirchoff stress-strain tensor S : R R (gradient of a convex internal/stored energy for hyper-elastic materials with nonlinear response)

3 viscoelastic materials exhibit time dependent stress-strain response when undergoing deformations : viscosity/relaxation approximations describe the (isothermal) motion in Lagrangian coordinates (with diffusive stress) which can be recast as Z ε = µ ε uε ( Z ε µ ε uε) = 1 t ε t ( Z ε S(u ε ) 1 ε exp( t s µ ) )( ε ε uε S(u ε ) (s) ds and the relaxation effects come from an (integral) memory-type term putting ε = 0 into the system, the equilibrium relation Z = S(u) + µ v x is formally recovered, that is interpreted as the passage from viscosity of the memory-type to viscosity of the rate-type B.D. Coleman, M.E. Gurtin, Thermodynamics with internal state variables, J. Chem. Physics (1967) C.M. Dafermos, Hyperbolic conservation laws in continuum physics, Fundamental Principles of Mathematical Sciences 325, Springer-Verlag, Berlin, 2000 A.E. Tzavaras, Materials with internal variables and relaxation to conservation laws, Arch. Ration. Mech. Anal. (1999) )

4 for visco-elastodynamics, the well-posedness in Sobolev spaces for large data is guaranteed by the results on weakly parabolic systems P. D Ancona, S. Spagnolo The Cauchy problem for weakly parabolic systems, Math. Ann. (1997) in one space dimension, the assumption that the nonlinear flux (stress-strain function) is globally Lipschitz, sup u R S (u) < +, stands for the sub-characteristic condition in multi-dimensional systems, a sufficient condition to compensate for the lack of parabolicity is a poly-convexity (or growth) assumption on the internal/stored energy C. Lattanzio, A.E. Tzavaras, Structural properties of stress relaxation and convergence from viscoelasticity to polyconvex elastodynamics, Arch. Ration. Mech. Anal. (2006) these are structural properties, whereas it should be proven for Navier-Stokes and Euler equations Y. Brenier, R. Natalini, M. Puel, On a relaxation approximation of the incompressible Navier-Stokes equations, Proc. Amer. Math. Soc. (2004)

5 features that make finite element methods computationally attractive : formal high-order accuracy for smooth solutions, nonlinear stability in curved (spatial) domains, to easily incorporate complex geometries, and support domain decomposition techniques for parallelization inherent implicit-explicit discretization of nonlinear terms difficulties to solve hyperbolic and convection-dominated problems : to choose the correct regularization mechanism, to capture discontinuities or strong gradients without producing spurious oscillations for relaxation approximations, the number of unknowns is increased relaxation + adaptive meshing provides a dissipative mechanism against the destabilizing effect of nonlinear response, combined with appropriate mesh refinement (by means of an error indicator) to detect the location of the singularities

6 standard semi-discrete finite element schemes let T h be (uniformly regular) partitions, with characteristic lenght h >0 (conforming) finite element spaces W h,k H0 1, k 1, of piecewise polynomials (globally continuous) on T h with consistency properties ( ) inf w h w L 2 + h w h w H 1 C h k w H k w h W h,k homogeneous/periodic boundary conditions finite element space W h,k 1 L 2 of piecewise discontinuous polynomials variational formulation : to find u ε h, v ε h, Z ε h W h,k solution to t u ε h, φ 1 + v ε h, x φ 1 = 0 t v ε h, φ 2 + Z ε h, x φ 2 = 0 ε t Z ε h, φ 3 + µ v ε h, x φ 3 = Z ε h, φ 3 + S(u ε h), φ 3 for all φ 1, φ 2, φ 3 W h,k (for the computational issues, semi-implicit discretizations in time are eventually considered, for which analogous estimates could also be proven)

7 for µ > ε, the system admits a positive definite symmetrizer µ/ε 0 1 H ε = 0 µ/ε which defines the hessian of a convex entropy-entropy flux pair (with dissipative behavior for the stiff/lower order source term) η(w ε ) = 1 (µ 2 ε 1)( u ε 2 + v ε 2) uε Z ε 2 q(w ε ) = ( µ ε 1) v ε Z ε with w ε = (u ε, v ε, Z ε ) T, in the sense that H ε w ε = w εη(w ε ) because of the symmetry, the mechanical energy is given by E ε (w ε ) = η(w ε ) dx = 1 2 Hε w ε, w ε = 1 2 wε, H ε w ε and it satisfies R d dt E ε (w ε (t)) = w ε t, w εη(w ε )

8 we use φ 1 = µ ε uε h Z ε h, φ2 =( µ ε 1)v ε h, φ3 =Z ε h uε h W h,k to obtain d dt E ε (w ε h) = 1 S(u ε ε h ) Zh ε, Zh ε uh ε revealing that the symmetrized form of the system is actually introduced to keep the conservation property for the finite element framework 1 ( ) S(u ε ε h)zh ε Zh ε 2 S(uh)u ε h ε + Zh ε uh ε dx R 1 ( 1 Zh ε 2 dx + 3 S(u ε ε 2 R 2 h ) 2 dx + 3 ) u ε R 2 h 2 dx R 1 2ε Z h ε 2 L + 3 ( 1 + sup S (ξ(u 2 2ε h)) ε ) 2 uh(t) ε 2 ξ(uh ε) W L 2 h under the generic assumption S(0)=0, and finally E ε T (w ε h) + 1 2ε T 0 Z ε h (t) 2 L 2dt E ε 0 (w ε h) + C 0 ε T 0 u ε h(t) 2 L 2dt for any T >0, with C 0 only depending upon the (global) Lipschitz condition for the nonlinear stress-strain function

9 for the method of modulated energy, the approximation of the relaxation variables is sought in lower-order finite element spaces we select φ 3 = x φ 2 W h,k 1 to obtain ε t Z ε h, x φ 2 µ x v ε h, x φ 2 = Z ε h, x φ 2 + S(u ε h), x φ 2 by differentiating the second equation with respect to time, we get tt v ε h, φ 2 + t Z ε h, x φ 2 = 0 and finally the following one-field equation, for any φ 2 W h,k t v ε h, φ 2 + S(u ε h), x φ 2 + µ x v ε h, x φ 2 + ε tt v ε h, φ 2 = 0 together with, for any φ 1 W h,k t u ε h, φ 1 + v ε h, x φ 1 = 0 so that the numerical system admits an equivalent (mixed) formulation, that is a standard finite element scheme perturbed by a wave operator (similar computation for fully-conforming finite element spaces)

10 we choose φ 1 =u ε h, φ2 =v ε h W h,k to obtain 1 d 2 dt uε h 2 L + v ε 2 h, x uh ε = 0, 1 d 2 dt v h ε 2 L + 2 S(uε h), x vh ε + µ x vh ε 2 L + ε d 2 dt tvh ε, vh ε L 2 ε t vh ε 2 L = 0, 2 taking advantage from the relaxation term, we deduce an estimate for the time derivative, and we modulate with a lower-order energy coming from the wave equation we fix λ > 1 and we choose φ 2 =ελ t v ε h W h,k to obtain ελ t vh ε 2 L + 2 ελ S(uε h), x t vh ε + µ ελ 1 d 2 dt xvh ε 2 L + 2 ε2 λ 1 d 2 dt tvh ε 2 L = 0 2 we sum together, rearranging the equalities as 1 d ) ( u ε 2 dt h 2 L + v ε 2 h + ε t vh ε 2 L + 2 ε2 (λ 1) t vh ε 2 L + µ ελ xv ε 2 h 2 L 2 + ε(λ 1) t vh ε 2 L + µ xv ε 2 h 2 L + 2 S(uε h) uh ε, x vh ε + ελ d dt S(uε h), x v ε h ελ t S(u ε h), x v ε h = 0

11 for the generic assumption S(0)=0, it holds S(uh) ε uh ε, x vh ε (1 + L S) 2 u ε µ h 2 L µ xvh ε 2 L 2 with L S >0 the Lipschitz constant of the stress-strain function using φ 1 =S (u ε h ) xv ε h W h,k 1 (with abuse of notation), we get that implies t u ε h, S (u ε h) x v ε h = x v ε h, S (u ε h) x v ε h µ x v ε h 2 L 2 ελ S (u ε h) t u ε h, x v ε h (µ ελl S ) x v ε h 2 L 2 0 under the sub-characteristic condition given by µ ελl S, which is satisfied for ε small enough in diffusive relaxation limits, but also essentially for a hyperbolic scaling thus, we conclude strong dissipative estimates for the modulated energy functional suggested by the relaxation terms (uniformly with respect to the relaxation and numerical parameters)

12 ????????? 1) implementation for multi-dimensional problems (including boundary conditions) and inverse inequalities for weak finite element spaces 2) to employ discontinuous finite elements to effectively handle shock waves arising in nonlinear elastic materials, for real applications 3) generalization to physical models with viscosity of memory-type leading to integro-differential operators (non-local with general kernels) 4) kinetic formulation of elastodynamics and related topics, as the relaxation method corresponds to some discrete kinetic approximation (equations of gas dynamics and magneto-hydrodynamics, shallow water equations, compressible Navier-Stokes equations with high Reynolds numbers, hydrodynamic models for semiconductor devices,...) 5) for the Euler equations, to devise an (incomplete) artificial/numerical viscosity with a (mesh dependent) higher order parameter, in view of an alternative approach to the resolution of contact discontinuities