Solutions of problems in computational physics. Waldemar Rachowicz Cracow University of Technology ul. Warszawska 24, Cracow, Poland

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1 Solutions of problems in computational physics Waldemar Rachowicz Cracow University of Technology ul. Warszawska 24, Cracow, Poland

2 Crack problem. Rvatshev s functions and BEM a) b) Figure: Zadanie szczeliny: a) przemieszczenia u z, b) naprȩżenia σ zz 19 March, 2018 Solutions of problems in computational physics 2 / 44

3 Crack problem a) b) Figure: Zadanie szczeliny: a) przemieszczenie u z, b) naprȩżenie σ zz wzd luż osi szczeliny 19 March, 2018 Solutions of problems in computational physics 3 / 44

4 Linear elasticity as a model elliptic bvp Formulation of linear elasticity: strong σ = f in Ω ɛ = 1 2 ( u + T u) σ = 2µɛ + λtrɛi u = û on Γ D σn = ˆt on Γ N 9 8 >= >< >; >: variational Find u V + û : a(u, v) = l(v) v V Z a(u, v) = [µɛ(u) : ɛ(v) + λtrɛ(u)trɛ(v)]dx Z Ω Z l(v) = f vdx + Ω ˆt n ds Γ N V = {v [H 1 (Ω)] 1 : v = 0 on Γ D }. Bilinear form a(u, v) is continuous and V -coercive, linear functional l(v) is continuous: M > 0 : a(u, v) M u 1,Ω v 1,Ω u, v H 1 (Ω) α > 0 : a(v, v) α v 2 1,Ω v V H1 (Ω) c > 0 : l(v) c v 1,Ω v H 1 (Ω) 19 March, 2018 Solutions of problems in computational physics 4 / 44

5 Aspects of adaptive methods: adaptivity Types: h-, p- and hp-adaptivity. Typical feed-back adaptive algorithm: 1. Solve the problem on the current mesh (99% of the cost). 2. Find error indicators η K 3. Stop if ( K η2 K )1/2 T OL 4. Refine elements K for which: η K > α max L η L, 0 < α < Go to 1. Goal oriented adaptivity: η K η u K ηg K 19 March, 2018 Solutions of problems in computational physics 5 / 44

6 Compressible Navier-Stokes equations Principles of conservation of mass, momentum and total energy: ϱ t + m i t [m j ] = 0 (0.1) x j + [ ] mi m j + δ ij p = [τ ij ] x j ϱ x j [ ] mj (e + p) = [ ] m i τ ij x j ϱ x j ϱ q i e t + Constitutive equations: p ϱθ = R, ι = C vϱθ, p = (γ 1)ι, γ = 1.4 τ ij = 2µε ij + λε kk, µ = µ 0 ( θ θ 0 ) 1.5 c 1 + θ 0 c 1 + θ, q i = κθ,i, κ = C p Pr µ, 19 March, 2018 Solutions of problems in computational physics 6 / 44

7 Flow around a plate, M = 3, Re = 1000 a Figure: Flow around a plate, M = 3, Re = (a) Distribution of density, (b) elevation of pressure b 19 March, 2018 Solutions of problems in computational physics 7 / 44

8 Flow around a cylinder, M = 8 a Figure: Flow around a cyliner, M = 8. (a) Distribution of density, (b) elevation of pressure b 19 March, 2018 Solutions of problems in computational physics 8 / 44

9 Flow around a cylinder with an impinging shock, M = 8 a Figure: Flow around a cyliner with an impinging shock, M = 8. (a) Distribution of density, (b) contour map of pressure 19 March, 2018 Solutions of problems in computational physics 9 / 44 b

(a) Distribution of pressure, (b) elevation of velocity u y 19

10 Incompressible Navier-Stokes equations { u t + (u )u + p = ν u, u = 0. a b Figure: Incompressible flow around a NACA-81 profile. (a) Distribution of pressure, (b) elevation of velocity u y 19 March, 2018 Solutions of problems in computational physics 10 / 44

11 Automatic h- or hp-adaptivity, L. Demkowicz, J. Kurtz, D. Pardo, W. Rachowicz Automatic h- or hp-adaptivity: We control two meshes: the coarse mesh h, p the fine mesh h/2, p + 1 Algorithm: 1. Investigate interpolation error of u h/2,p+1 on the elements of the coarse mesh with various trial refinements. 2. Select refinements resulting in the largest reduction of interpolation error per 1 new dof. 3. Perform selected refinements on the coarse mesh. 4. Obtain the fine mesh. 5. Solve on fine mesh (99% of the cost). 6. Go to 1. Goal oriented version of the algorithm exists. 19 March, 2018 Solutions of problems in computational physics 11 / 44

12 Cantilever beam Figure: Cantilever beam. Coarse and fine hp meshes, distribution of σ yz, σ 0. Ω = [0, 1] [0, 4] [0, 1], load: t z = 1 for 3.5 < y < March, 2018 Solutions of problems in computational physics 12 / 44

13 Hyperboloidal shell Figure: Hyperboloidal shell. Coarse and fine hp meshes; σ φφ, σ zz, σ 0. Ω : r2 z2 = 1, z [ 3, 1], t = 0.05; load: t x = 1 19 March, 2018 Solutions of problems in computational physics 13 / 44

14 Streamer - cable for aquisition of seismic data a b Figure: (a) structure of a segment (gel, skin, spacer, rope, hydrophone) (b) h-adaptive mesh 19 March, 2018 Solutions of problems in computational physics 14 / 44

15 Streamer Time-harmonic elasto-dynamics: σ ρω 2 u = 0 in Ω u n n = iωρ wc w u n on Γ N u = û on Γ D hydrophone E = 4.0 GPa, ν = 0.4, ρ = 1500 kg/m 3 gel E =.24 GPa, ν = 0.43, ρ = 1040 kg/m 3 skin E = 0.02 GPa, ν = 0.45, ρ = 1200 kg/m 3, spacer E = 1.8 GPa, ν = 0.3, ρ = 1200 kg/m 3, rope E = 41 GPa, ν = 0.3, ρ = 1400 kg/m 3 19 March, 2018 Solutions of problems in computational physics 15 / 44

16 Streamer a b c Figure: Distribution of pressure [db] (a) over a plane of symmetry, (b) over a hydrophone, (c) along a center line (h and hp simulations) 19 March, 2018 Solutions of problems in computational physics 16 / 44

17 Goal-oriented adaptivity Hyperboloidal shell: r 2 a 2 z2 c 2 = 1, a in = 1, a out = 1.05, c in = c out = 2, z [ 3, 1] Quantities of interest: F (u) = wσ ij ζ k dv, D( S) i, j - shell local directions S - small part of middle surface, meas(w) = 1 i, j = 1, k = 1: F = (m 11 ) av - bending moment i, j = 1, k = 0: F = (n 11 ) av - membrane force Tangential orders p 4 to avoid locking! 19 March, 2018 Solutions of problems in computational physics 17 / 44

(d,e,f) for n 11. Results: (m 11 ) av = 0.151 ± 0.7%, (n 11 ) av = 5.

18 Hyperboloidal shell, goal-oriented adaptivity a b c d e f Figure: (a,b,c) coarse and fine mesh, u r of Green s function for m 11 ; (d,e,f) for n 11. Results: (m 11 ) av = ± 0.7%, (n 11 ) av = 5.03 ± 0.06% 19 March, 2018 Solutions of problems in computational physics 18 / 44

19 Cantilever beam - goal-oriented adaptivity a b c d Figure: (a,b) an h- and hp-adaptive meshes, (c,d) u x and σ xx for Green s function. Q.o.i. F = ω σ 11dx, ω = [0.9375, 1.0] 3. Result: F = ± 0.03% (h-adapt.), F = ± 0.02% (hp-adapt.) 19 March, 2018 Solutions of problems in computational physics 19 / 44

20 BVP in electromagnetics Time-harmonic Maxwell equations: E = jωµh, H = 1 jωµ E, H = jωɛe + σe + J imp, ˆɛ = ɛ jσ/ω Boundary-value problem: 1 / µ E ω2ˆɛe = jωj imp in Ω, F, n E = M imp S on Γ D, n E = jωj imp S on Γ N, Variational formulation: Find E such that n E = M imp S on Γ D and: Z Z 1 Ω µ E F dx ω 2ˆɛE Z Z F dx = jω J imp F ds + jω Ω Ω {z } a(e,f ) Ω J imp S Γ N F ds, {z } l(f ) F : F t = 0, on Γ D. 19 March, 2018 Solutions of problems in computational physics 20 / 44

21 Time-harmonic Maxwell equations, FEM a b Figure: Rozpraszanie fal p laskich na wnȩce, λ = 0.5. Pole elektryczne E y : a) czȩść rzeczywista, b) czȩść urojona 19 March, 2018 Solutions of problems in computational physics 21 / 44

22 Time-harmonic Maxwell equations, FEM Figure: Rozpraszanie fal p laskich na wnȩce, λ = 0.5. Przekroj na rozpraszanie radarowe (RCS) 19 March, 2018 Solutions of problems in computational physics 22 / 44

23 Time-harmonic Maxwell equations, BEM + FEM We solve the Stratton-Chu integral equation: Z Z Z «E(y) = y [γ + D k (E) G ds γ + N (E) G ds + 1 y Γ [γ + N Γ Γ k (E)]G ds. Γ γ + D E := E n, γ+ N := k 1 ( E) n, G(x y) = e jkr, r = x y. 4πr a b Figure: Rozpraszanie na kuli przewodz acej. Sk ladowa Re(E y ) rozwi azania: a) czȩść rzeczywista b) czȩść urojona 19 March, 2018 Solutions of problems in computational physics 23 / 44

24 Time-harmonic linear acoustics, BEM + FEM Burton-Miller formulation, α [0, 1], α = (1 α)j/k: Z» α α p p(y) + (y) = α p G p «G + α 2 G p p «G ds, 2 2 n y Γ n x n x n y n x n x n y G(x y) = ejkr, r = x y. 4πr a b Figure: Rozpraszanie na walcowej pow loce sprȩżystej - ciśnienie: a) czȩść rzeczywista, b) czȩść urojona 19 March, 2018 Solutions of problems in computational physics 24 / 44

25 Inverse medium scattering problems in electromagnetics Figure: Transmitters and receivers surrounding the scatterer 19 March, 2018 Solutions of problems in computational physics 25 / 44

26 Inverse medium scattering in electromagnetics Inverse medium scattering: Consider incident waves E in m, m = 1,..., M For each of waves E in m we measure the scattered waves E mn at N observation points, x n, n = 1,..., N. Based on the measurements E mn we wish to evaluate the distribution of ˆɛ (x) by matching the measurements with the numerical simulations for trial ˆɛ(x) { } ˆɛ (x) = arg min E mn (ˆɛ) E mn 2 + M(ˆɛ) m,n where M is a (possible) stabilizing functional. 19 March, 2018 Solutions of problems in computational physics 26 / 44

Ex. kidney-like shape, complex, complete data a) b) Figure: Reconstruction of complex-valued ˆɛ(x) with complete measurement data, kidney-like scatterer

27 Ex. kidney-like shape, complex, complete data a) b) Figure: Reconstruction of complex-valued ˆɛ(x) with complete measurement data, kidney-like scatterer with a tumor: a) exact Re(ˆɛ), b) reconstructed Re(ˆɛ), data E mn perturbed by random noise 19 March, 2018 Solutions of problems in computational physics 27 / 44

Ex. kidney-like shape, complex, complete data c) d) Figure: Reconstruction of complex-valued ˆɛ(x) with complete measurement data, kidney-like scatterer

28 Ex. kidney-like shape, complex, complete data c) d) Figure: Reconstruction of complex-valued ˆɛ(x) with complete measurement data, kidney-like scatterer with a tumor: c) exact Im(ˆɛ), d) reconstructed Im(ˆɛ); data E mn perturbed by random noise 19 March, 2018 Solutions of problems in computational physics 28 / 44

tumor in two subspaces: a)meshes, b) exact Re(ˆɛ), c) reconstructed Re(ˆɛ), data E mn

29 Ex. scatterer in two half-spaces, complex-valued ˆɛ Figure: Reconstruction of complex-valued ˆɛ(x) with incomplete measurement data, kidney-like scatterer with a tumor in two subspaces: a)meshes, b) exact Re(ˆɛ), c) reconstructed Re(ˆɛ), data E mn perturbed by random noise 19 March, 2018 Solutions of problems in computational physics 29 / 44 a) b) c)

tumor in two subspaces: a) exact Im(ˆɛ), b) reconstructed Im(ˆɛ), E mn perturbed by

30 Ex. scatterer in two half-spaces, complex-valued ˆɛ a) b) Figure: Reconstruction of complex-valued ˆɛ(x) with incomplete measurement data, kidney-like scatterer with a tumor in two subspaces: a) exact Im(ˆɛ), b) reconstructed Im(ˆɛ), E mn perturbed by random noise 19 March, 2018 Solutions of problems in computational physics 30 / 44

Limb-like scatterer in a cylinder, complex-valued ˆɛ a) b) c) Figure: Reconstruction of complex-valued ˆɛ(x) with incomplete measurement data, limb-like scatterer

31 Limb-like scatterer in a cylinder, complex-valued ˆɛ a) b) c) Figure: Reconstruction of complex-valued ˆɛ(x) with incomplete measurement data, limb-like scatterer with tumors: a) meshes, b) exact Re(ˆɛ), c) reconstructed Re(ˆɛ), data E mn perturbed by random noise 19 March, 2018 Solutions of problems in computational physics 31 / 44

limb-like scatterer with tumors: a) exact Im(ˆɛ), b) reconstructed Im(ˆɛ), data

32 Limb-like scatterer in a cylinder, complex-valued ˆɛ a) b) Figure: Reconstruction of complex-valued ˆɛ(x) with incomplete measurement data, limb-like scatterer with tumors: a) exact Im(ˆɛ), b) reconstructed Im(ˆɛ), data E mn perturbed by random noise 19 March, 2018 Solutions of problems in computational physics 32 / 44

33 Ex. 2 PEC spheres, r = 0.555λ, Linear Sampling Method (Peter Monk) Figure: Inverse scattering recovery: two PEC spheres 19 March, 2018 Solutions of problems in computational physics 33 / 44

34 Example: two PEC hexahedra, a = 1.12λ, Linear Sampling Method Figure: Inverse scattering recovery: two PEC hexahedra 19 March, 2018 Solutions of problems in computational physics 34 / 44

35 Finite anisotropic elasticity + contact + plasticity. The notions of finite elasticity: x(x, t) - Lagrangian description F = X x - deformation gradient C = F T F - right Cauchy Green tensor J = det(c) 1/2 = det(f ) > 0 - volume ratio Multiplicative decomposition of deformation gradient: F = J 1/3 F, C = J 2/3 C Fields of fibre directions: M i (X), i = 1, 2, M i = 1. Structural tensors: A i = M i M i, i = 1, 2 Invariants: I 1 = tr(c), I 2 = 1 2 [(tr(c)2 tr(c 2 ), I 4i = C : A i, Ī 1 = J 2/3 I 1, Ī 2 = J 1/3 I 2, Ī 4i = J 2/3 I 4i. 19 March, 2018 Solutions of problems in computational physics 35 / 44

36 Formulation of finite elasticity bvp Split of strain energy function (R. Flory): 2 Ψ = Ψ vol (J) + Ψ iso (Ī 1, Ī 2 ) + W fi (Ī 4i ) i=1 The second Piola-Kirchhoff stress tensor: S = 2 Ψ C = pjc 1 + 2J 2/3 {Ψ iso,1 Dev[I] + Ψ iso,2 Dev[ C 2 ] + 2 i=1 W fi,4dev[a i ]} Strain energies: Ψ vol = K 4 (J 1)2, Ψ iso = µ 2 (Ī 1 3), Ψ fi = { k 1i 2k 2i exp [k2i (Ī 4i 1) 2 ] 1 }. G. Holzapfel, T.Gasser, R. Ogden HGO 19 March, 2018 Solutions of problems in computational physics 36 / 44

Figure: Contact of two elastic cylinders solved on

meshes, c) the mean pressure, and d) effective

37 Figure: Contact of two elastic cylinders solved on meshes of orders p = 1, p = 2 and p = 4 (in subsequent rows). The columns show: a) undeformed meshes, b) deformed meshes, c) the mean pressure, and d) effective stress σ 0 19 March, 2018 Solutions of problems in computational physics 37 / 44 p=1 p=2 p=4 a b c d

the reference configuration, b,e) displacements u z.

38 a b c d e f Figure: Half-cylindrical die intruding into elastic block and a thin-walled tube pressed by a flexible plate: a,d) an h-adaptive mesh on the reference configuration, b,e) displacements u z. c,f) effective stress σ 0 19 March, 2018 Solutions of problems in computational physics 38 / 44

39 The idea of angioplasty a b c Figure: Angioplasty: a) artery, stent and catheter, and displacements u r on (b) the reference and (c) the deformed configuration 19 March, 2018 Solutions of problems in computational physics 39 / 44

40 Angioplasty a Figure: Angioplasty. Distribution of: (a) pressure, and (b) stresses σ rφ in the vicinity of the contact of the tube b 19 March, 2018 Solutions of problems in computational physics 40 / 44

41 Angioplasty r r Location of artery surface Location of artery surface 2 2 1,8 1,8 1,6 1,6 1,4 1,4 1,2 1 0,8 before loading after loading after unloading 1,2 1 0,8 before loading after loading after unloading 0,6 0,6 0,4 0, a z 0,4 0, b z Figure: Location of internal surface of the tube at different stages of loading process for tubes types (a) and (b) 19 March, 2018 Solutions of problems in computational physics 41 / 44

on the artery, and on the balloon (c) 19

42 Angioplasty, a quadratic mesh a b c Figure: Angioplasty. Approximation on a quadratic mesh: (a) the mesh, (b) radial displacements u r on the artery, and on the balloon (c) 19 March, 2018 Solutions of problems in computational physics 42 / 44

Mixed FEM formulation for highly anisotropic elastic

.. (with Adam Zdunek) a Figure: c b d Pressurization

mesh, (c) effective stress devσ 0 on a uniform p = 4

43 Mixed FEM formulation for highly anisotropic elastic materials (with Adam Zdunek) a Figure: c b d Pressurization od a 2-layer toroidal tube (the data correspond to a rabbit carotid artery). (a) An initial p = 2 mesh, (b) detail of an h-adaptive mesh, (c) effective stress devσ 0 on a uniform p = 4 mesh, (d) effective stress on a detail of an h-adaptive mesh 19 March, 2018 Solutions of problems in computational physics 43 / 44

44 Dziȩkujȩ! 19 March, 2018 Solutions of problems in computational physics 44 / 44

NONLINEAR CONTINUUM FORMULATIONS CONTENTS

NONLINEAR CONTINUUM FORMULATIONS CONTENTS Introduction to nonlinear continuum mechanics Descriptions of motion Measures of stresses and strains Updated and Total Lagrangian formulations Continuum shell