Tectonic evolution at mid-ocean ridges: geodynamics and numerical modeling. Second HPC-GA Workshop

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1 .. Tectonic evolution at mid-ocean ridges: geodynamics and numerical modeling. Second HPC-GA Workshop Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Politecnico di Milano - MOX, Dipartimento di Matematica F. Brioschi Bilbao, March 11th 2013 M X MILANO MODELING AND SCIENTIFIC COMPUTING MODELLISTICA E CALCOLO SCIENTIFICO POLITECNICO Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 1/ 55

2 Aims and motivations Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 2/ 55

3 Aims and motivations Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 3/ 55

4 Continental rift Analogue models The analogue models have the following disadvantages: they lack the thermo-mechanical description of phenomena, Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 4/ 55

5 Continental rift Analogue models The analogue models have the following disadvantages: they lack the thermo-mechanical description of phenomena, they are very expensive to setup and perform, Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 4/ 55

6 Continental rift Analogue models The analogue models have the following disadvantages: they lack the thermo-mechanical description of phenomena, they are very expensive to setup and perform, they are not general, but specific for a given region, Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 4/ 55

Continental rift Analogue models The analogue models have the following disadvantages: they lack the thermo-mechanical description of phenomena, they are very expensive to setup

7 Continental rift Analogue models The analogue models have the following disadvantages: they lack the thermo-mechanical description of phenomena, they are very expensive to setup and perform, they are not general, but specific for a given region, they are not replicable. Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 4/ 55

8 Aims and motivations the development of the mathematical and numerical model of mid-oceanic ridges: Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 5/ 55

9 Aims and motivations the development of the mathematical and numerical model of mid-oceanic ridges: improvement of the boundary conditions to model the ridge migration; a better thermo-mechanical model (melting); Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 5/ 55

10 Aims and motivations the development of the mathematical and numerical model of mid-oceanic ridges: improvement of the boundary conditions to model the ridge migration; a better thermo-mechanical model (melting); development of the mathematical tools for the defition of a numerical sandbox useful to reproduce the continental-rift evolution; Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 5/ 55

11 Aims and motivations the development of the mathematical and numerical model of mid-oceanic ridges: improvement of the boundary conditions to model the ridge migration; a better thermo-mechanical model (melting); development of the mathematical tools for the defition of a numerical sandbox useful to reproduce the continental-rift evolution; development and application of meshfree methods and variational integrators for the simulation of the numerical sandbox. Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 5/ 55

Mid-ocean ridges Mathematical model We can treat the litosphere and asthenosphere as higly viscous incompressible fluid div(2ηd) p + ρg = 0 div v = 0 θ + v θ = div (κ θ) t η = η 0 e Cθ The

12 Mid-ocean ridges Mathematical model We can treat the litosphere and asthenosphere as higly viscous incompressible fluid div(2ηd) p + ρg = 0 div v = 0 θ + v θ = div (κ θ) t η = η 0 e Cθ The Frank-Kamenetskii linearization is use to approximate the viscosity law η = 1 2A ( µ I 2 ) n 1 ( ) h m e E+pV RT b Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 6/ 55

13 Mid-ocean ridges The initial condition for the temperature field is the mean oceanic geotherm ( ) θ θ 0 y = erf θ M θ 0 2 κτ Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 7/ 55

14 Mid-ocean ridges Results of numerical simulations in a steady-state regime. Results of numerical simulations with ridge migration in act. Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 8/ 55

15 Mid-ocean ridges Boundary conditions The previous boundary conditions have to be improved, since they do not depend upon the rheology, (in particular on the viscosity); the simulation showed they can lead to incorrect results near boundaries. Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 9/ 55

16 Mid-ocean ridges Boundary conditions A different strategy has been developed; these are the main hypothesis: the boundaries are far enough from the ridge, the upwelling motion is negligible compared to shearing. y v extension. 0 x v migration h y dq u(y) = u 0 + s 0 η(q) p(y) = p 0 + y 0 f(q) dq Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 10/ 55

17 Mid-ocean ridges: Mathematical model div(2ηɛ(u)) p + f = 0 divu = 0 div(k T) + u T = 0 in Ω in Ω in Ω u = ū on Γ 1 ((2ηɛ(u) pi)n = g on Γ 2 where and T = T on Γ 3 k T n = s on Γ 4 Γ 1 Γ 2 =, Γ 3 Γ 4 =, Γ 1 Γ 2 = Ω, Γ 3 Γ 4 = Ω, Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 11/ 55

18 Mid-ocean ridges: Numerical Discretization (1) A(T) B T 0 u f B 0 0 p = C(u) T g expanding the nonlinear terms at the first order we get Newton s method A(T) B T A (T) δu f A(T) B T 0 u B 0 0 δp = 0 B 0 0 p C (u) 0 C(u) δt g 0 0 C(u) T Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 12/ 55

19 Mid-ocean ridges: Numerical Discretization (1) The Newton step can be solved using the block LU decomposition, denoting [ ] A(T) B T A = B B 0 1 = [ A (T) 0 ] B 2 = [ C (T) 0 ], so we get the following linear system [ ] [ I 0 A B T 1 B 2 A 1 I 0 S where S is the Schur complement, se we need a good Stokes solver, a good preconditioner for S ] [ ] δx = δy S = C B 2 A 1 B T 1 [ ] f g Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 13/ 55

20 Mid-ocean ridges: Preconditioners the Stokes problem is solved using the Schur complement; Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 14/ 55

21 Mid-ocean ridges: Preconditioners the Stokes problem is solved using the Schur complement; the preconditioners for A and the (Stokes-)Schur complement are, respectively, the algebraic multigrid (AMG) preconditioner (Trilinos); Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 14/ 55

22 Mid-ocean ridges: Preconditioners the Stokes problem is solved using the Schur complement; the preconditioners for A and the (Stokes-)Schur complement are, respectively, the algebraic multigrid (AMG) preconditioner (Trilinos); the viscosity-scaled pressure mass-matrix. Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 14/ 55

23 Mid-ocean ridges: Preconditioners the Stokes problem is solved using the Schur complement; the preconditioners for A and the (Stokes-)Schur complement are, respectively, the algebraic multigrid (AMG) preconditioner (Trilinos); the viscosity-scaled pressure mass-matrix. Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 14/ 55

24 Mid-ocean ridges: Preconditioners the Stokes problem is solved using the Schur complement; the preconditioners for A and the (Stokes-)Schur complement are, respectively, the algebraic multigrid (AMG) preconditioner (Trilinos); the viscosity-scaled pressure mass-matrix. these preconditioners are almost optimal. Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 14/ 55

25 Mid-ocean ridges: Preconditioners the Stokes problem is solved using the Schur complement; the preconditioners for A and the (Stokes-)Schur complement are, respectively, the algebraic multigrid (AMG) preconditioner (Trilinos); the viscosity-scaled pressure mass-matrix. these preconditioners are almost optimal. The matrix S is preconditioned with the iterative inverse of C (this last problem is preconditioned with AMG preconditioner). Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 14/ 55

26 Mid-ocean ridges: Preconditioners the Stokes problem is solved using the Schur complement; the preconditioners for A and the (Stokes-)Schur complement are, respectively, the algebraic multigrid (AMG) preconditioner (Trilinos); the viscosity-scaled pressure mass-matrix. these preconditioners are almost optimal. The matrix S is preconditioned with the iterative inverse of C (this last problem is preconditioned with AMG preconditioner). this preconditioner seems to scale well with the number of degrees of freedom (for each numerical experiments we get about iterations to solve the temperature). Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 14/ 55

27 Mid-ocean ridges: Numerical treatment of stress BC To compute the stress boundary conditions we use the fixed point iteration method, with the following scheme given a solution (u, p, T) the boundary condition g is computed from g a new solution (u, p, T) is computed Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 15/ 55

28 Mid-ocean ridges Results of numerical simulations in a steady-state regime. Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 16/ 55

29 Mid-ocean ridges Results of numerical simulations with migration Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 17/ 55

30 Mid-ocean ridges Scalability results: dof Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 18/ 55

31 Mid-ocean ridges Scalability results: dof Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 19/ 55

32 Mid-ocean ridges Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 20/ 55

33 Mid-ocean ridges Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 21/ 55

34 Mid-ocean ridges Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 22/ 55

35 Geomod example Figure : Extension: sylicon on all the base Figure : Sylicon only in the central part Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 23/ 55

36 Drucker Prager: Conf. 1 (MI, SI, V) Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 24/ 55

37 Drucker Prager: Conf. 2 (MI, SI, V) Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 25/ 55

38 Von Mises: Conf. 1 (MI, SI, V) Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 26/ 55

39 Von Mises: Conf. 2 (MI, SI, V) Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 27/ 55

40 Continental rift Mathematical model The previous model is extended to include the elastic behavior of lithosphere (upper-convected Maxwell model) div T p + ρg = 0 div v = 0 T + λt = 2ηD ρ dh dt = dp + T : D + ρr div q dt from the dimensional analysis we get the following estimates for the upwelling ρ dh dt dp dt ρr ρ dh dt div q ρ dh dt Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 28/ 55

41 Optimal transportation mesh-free method Continuum mechanic problems which involve large deformations or which are formulated with respect to the reference configuration need a continuous remeshing or some other strategies to overcome the difficulties arising from the continuous mesh update. Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 29/ 55

42 Optimal transportation mesh-free method Continuum mechanic problems which involve large deformations or which are formulated with respect to the reference configuration need a continuous remeshing or some other strategies to overcome the difficulties arising from the continuous mesh update. Mesh-free methods are an alternative, since they give up the mesh and use only a point set to discretize the problem. Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 29/ 55

43 Optimal transportation mesh-free method Continuum mechanic problems which involve large deformations or which are formulated with respect to the reference configuration need a continuous remeshing or some other strategies to overcome the difficulties arising from the continuous mesh update. Mesh-free methods are an alternative, since they give up the mesh and use only a point set to discretize the problem. the OTM is designed for the continuum mechanic problems, since it inherits from the continuous problem the symmetries and conservation properties, avoiding some issues of other mesh-free methods. Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 29/ 55

44 Local maximum-entropy shape functions Set of nodes X = {x a R d, a = 1,..., N} Non-negative shape functions p a : conv X R 0 First order consistency conditions p a (x) = 1 a p a (x)x a = x a Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 30/ 55

45 Local maximum-entropy shape functions. Local maximum-entropy problem. For x fixed, minimize βu x (p) H(p) subject to p a 0 p a = 1 a p a x a = x a. where U x (p) = a p a x x a 2 and H(p) = a p a log p a Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 31/ 55

46 Local maximum-entropy shape functions. Local maximum-entropy problem. Defined the partition function Z(x, λ) = a e β x x a 2 +λ (x x a ). For β(x) R 0 and x int conv X. Then the unique solution of the problem is p a (x) = where λ is the unique solution of 1 Z(x, λ ) e β x x a 2 +λ (x x a ),. λ = arg min λ). λ RdZ(x, Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 32/ 55

47 Local maximum-entropy shape functions Spatial smoothness β C r = p a C r Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 33/ 55

48 Local maximum-entropy shape functions Spatial smoothness β C r = p a C r Smoothness with respect β p(β) C 0 ([0, )), p(β) C ((0, )) Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 33/ 55

49 Local maximum-entropy shape functions Spatial smoothness β C r = p a C r Smoothness with respect β p(β) C 0 ([0, )), p(β) C ((0, )) Limits of shape functions as β p a converge to a Delaunay convex approximants as β Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 33/ 55

50 Local maximum-entropy shape functions Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 34/ 55

51 Convergence Let us consider the following problem Ω = ( 1, 1)x( 1, 1). γ = βh 2 u = 2 x 2 y 2, in Ω u = 0 on Ω Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 35/ 55

52 Convergence: L 2 norm Nodes Relative error γ L 2 error Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 36/ 55

53 Convergence: H 1 norm Relative error H 1 error γ Nodes Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 37/ 55

54 Optimal transportation The motion of an inviscid fluid of non-interacting particles in R d is governed by the equations { ρ + div (ρu) = 0 (ρu) + div (ρu u) = 0 This problem can be recasted as an optimal transportation problem, that admits a variational formulation minimize b 1 J(ρ, u) = a R d 2 ρ u 2 dx dt subject to ρ + div (ρu) = 0 Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 38/ 55

55 Optimal transportation The problem can be recasted in term of deformation map φ hence the velocity and the density are given by u(x, t) = φ t (φ 1 t (x)) ρ(x, t) = ρ a (φ 1 t (x)) det φ t (φ 1 t and Benamou and Brenier (1999) showed that 1 inf J(ρ, u) = inf ρ a (x) φ b (x) x 2 dx b a R d (x)) Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 39/ 55

56 Optimal transportation - discretization Given a finite set of timestep t 0 = a, t 1, t 2,..., t N = b, the semidiscrete problem is given by J = k 1 t k+1 t k 1 2 ρ k(x) φ k k+1 (x) x 2 dx, then using a quadrature rule J = k n 1 1 t k+1 t k 2 ρ k(x n,k )v n φ k k+1 (x n,k ) x n,k 2 denoting x n,k+1 = φ k k+1 (x n,k ) and m n,k = ρ k (x n,k )v n, then the solution of this problem is x n,k+1 = x n,k + (t k+1 t k ) x n,k x n,k 1 t k t k 1 m n,k = m n,0 Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 40/ 55

57 Optimal transportation - solid and fluid flows This framework can be extended for inviscid fluids and elastic solids, adding a free energy term to the functional J J(ρ, u, C) = b a R d 1 2 ρ u 2 ρu(ρ, C) dx dt The kinetic energy term is discretized in same way, but for the free energy the trapezoidal quadrature rule is used J = 1 1 t k+1 t k 2 ρ k φ k k+1 (x) x 2 dx+ k + t [ ] k+1 t k ρ k U(ρ k, C k ) dx + ρ k+1 U(ρ k+1, C k+1 ) dx 2 Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 41/ 55

58 Optimal transportation - solid and fluid flows Two points sets are introduced: nodal set {x a }, used to construct the shape functions material set {x n }, used as quadrature points So the displacement map can be written as a linear combination of shape functions φ k k+1 (x) x = a p a (x)d a, and the problem can be completely discretized J = k n + t k+1 t k 2 1 t k+1 t k 1 2 m n,k φ k k+1 (x n,k ) x n,k 2 + [m n,k U(ρ n,k, C n,k ) + m n,k+1 U(ρ n,k+1, C n,k+1 )] Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 42/ 55

59 Optimal transportation - solid and fluid flows The solution of this problem is ( d k = (t k+1 t k )M 1 k l k + t ) k+1 t k 1 f k 2 x n,k+1 = φ k k+1 (x n,k ), x a,k+1 = φ k k+1 (x a,k ) v n,k+1 = v n,k det φ k k+1 (x a,k ) where l a,k = n m n,k x n,k x n,k 1 t k t k 1 p a,k (x n,k ) M k,ab = n m n,k p a,k (x n,k )p b,k (x n,k )I f a,k = n [ρ n,k b n,k p a,k (x n,k ) + σ n,k : p a,k (x n,k )] v n,k Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 43/ 55

60 Application to the Stokes flows ( ) inf u 2 p div u dx λ (u ū) dσ (u,p) Ω Ω Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 44/ 55

61 Optimal transportation - solid and fluid flows U = λ 2 [tre]2 + µtre 2 λ = Pa µ = Pa Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 45/ 55

62 Veselov-type discretization Choosen the discrete set of times 0 = t 0 < t 1 < t 2 < < t N 1 < t N = T then the motion is described by a sequence of positions q 0, q 1, q 2,..., q N 1, q N and the discrete action is defined by the sum S q = N 1 k=0 L d (q k, q k+1 ) where L d (q k, q k+1 ) is called discrete Lagrangian, it depends on two subsequent positions and it s a reasonable approximation of the action between the configurations q k and q k+1 L d (q k, q k+1 ) tk+1 t k L(q, q) dt Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 46/ 55

63 Variational Integrators in time: Störmer-Verlet integration Using the trapezoidal quadrature rule and approximating the velocity with finite difference we get the discrete Lagrangian [ ( ) ] 1 qk+1 q 2 k L d (q k, q k+1 ) = (t k+1 t k ) V(q k) + V(q k+1 ) 2 t k+1 t k 2 This leads to the well known Störmer-Verlet method q k+1 q k t k+1 t k q k q k 1 t k t k 1 = t k+1 t k 1 F(q k ) 2 and the related velocity Verlet method q k+1 q k = p k + t k+1 t k F(q k ) t k+1 t k 2 p k+1 p k = F(q k) + F(q k+1 ) t k+1 t k 2 Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 47/ 55

64 Midpoint rule Instead, using the midpoint quadrature rule we get the following discrete Lagrangian [ ( ) 1 qk+1 q 2 ( ) ] k qk + q k+1 L d (q k, q k+1 ) = (t k+1 t k ) V 2 t k+1 t k 2 and the related method is q k+1 q k = p k + t ( ) k+1 t k qk+1 + q k F t k+1 t k 2 2 ( ) p k+1 p k qk+1 + q k = F t k+1 t k 2 Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 48/ 55

65 Other variational integrators Most of standard symplectic methods can be written as variational integrators, among them: Newmark-β methods, the first proof of symplecticity of these methods exploits the framework of variational integrators, [ ( 1 η β (q 1 ) η β ) 2 (q 0 ) L d (q 0, q 1 ) = h V(η β (q 0 ))] 2 h symplectic partitioned Runge-Kutta Galerkin methods (Lobatto IIIA-IIIB) L d (q 0, q 1 ) ext {S NI (q) : q V N, q(0) = q 0, q(h) = q 1 } Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 49/ 55

66 Wave equation Probably the simplest model is the linear wave equation, which represents a linearization of the hyperelastic problem. The related Lagrangian density in the one dimensional space is [ ( φ L = 1 ) 2 ( c φ ) ] 2 2 t x Applying the trapezoidal quadrature rule in time we get the semi-discrete Störmer-Verlet scheme φ k+1 φ k ψ dx = p k ψ t k+1 t k c 2 φ k ψ B t k+1 t k B 2 x x dx B p k+1 p k ψ dx = c 2 t k+1 t k B x ( φk + φ k+1 2 ) ψ x dx Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 50/ 55

67 Wave equation c = 1; P 1 ; t = 0.01s; I = [0, 2π]; N x = 100. uh u log 2 2 u t position momentum Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 51/ 55

68 Wave equation: linear momentum conservation (=0) p dx log 2 2π t Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 52/ 55

69 Wave equation: energy Eh E E t Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 53/ 55

70 Conclusions and further developments improvement of BC treatment; preconditioners; preliminary results on OTM. Future developments of this project introduction of the melting; development of a fluid-structure and structure-structure method based on OTM; application of OTM and VI to geodynamic problems, coupling the mechanical problem with the thermodynamic. Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 54/ 55

71 Thank you for your attention! Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 55/ 55

Well-balanced DG scheme for Euler equations with gravity

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