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Isogeometric mortar methods with applications in contact mechanics PhD dissertation defense Brivadis Ericka Supervisor: Annalisa Buffa Doctor of Philosophy in Computational Mechanics and Advanced Materials, XXVIII Cycle (2012-2015) Pavia, January 19th, 2017

Context of the work This research work is part of a project between the of Pavia and the company which aims to treat complex contact problems with isogeometric analysis. It has been done in collaboration with A. Buffa ( Pavia), G. Elber ( ), M. Martinelli ( Pavia), F. Massarwi ( ), L. Wunderlich ( ) and B. Wohlmuth ( ). Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 2 / 45

Outline 1 IGA basics 2 Isogeometric mortar methods 3 Applications in contact mechanics Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 3 / 45

IGA basics Univariate B-Splines The parametric unit interval 0 1 Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 4 / 45

IGA basics Univariate B-Splines The parametric unit interval is split by the breakpoints ζ j (j = 1,..., N) leading to a parametric mesh. 0 1 2 3 4 1 5 5 5 5 U = {0, 1 5, 2 5, 3 5, 4 5, 1} Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 4 / 45

IGA basics Univariate B-Splines The parametric unit interval is split by the breakpoints ζ j (j = 1,..., N) leading to a parametric mesh. 0 1 2 3 4 1 5 5 5 5 U = {0, 1 5, 2 5, 3 5, 4 5, 1} Each breakpoint ζ j has a multiplicity m j. The breakpoints and their multiplicities define the knot vector Ξ = {ζ 1,..., ζ }{{} 1, ζ 2,..., ζ 2,..., ζ }{{} N,..., ζ N }. }{{} m 1 times m 2 times m N times Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 4 / 45

IGA basics Univariate B-Splines The parametric unit interval is split by the breakpoints ζ j (j = 1,..., N) leading to a parametric mesh. 0 1 2 3 4 1 5 5 5 5 U = {0, 1 5, 2 5, 3 5, 4 5, 1} Each breakpoint ζ j has a multiplicity m j. The breakpoints and their multiplicities define the knot vector Ξ = {ζ 1,..., ζ }{{} 1, ζ 2,..., ζ 2,..., ζ }{{} N,..., ζ N }. }{{} m 1 times m 2 times m N times Given a degree p and a knot vector Ξ, the univariate (p + 1)-order B-Splines can be defined recursively. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 4 / 45

IGA basics Univariate B-Splines Important properties 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Univariate quadratique B-Spline functions defined on the knot vector Ξ = {0, 0, 0, 1 5, 2 5, 2 5, 3 5, 4 5, 1, 1, 1}. 0 1 2 3 4 1 5 5 5 5 We note the C 0 continuity in ζ j = 2 5. Each B p i is a piecewise positive polynomial of degree p and has a local support. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 5 / 45

IGA basics Univariate B-Splines Important properties 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Univariate quadratique B-Spline functions defined on the knot vector Ξ = {0, 0, 0, 1 5, 2 5, 2 5, 3 5, 4 5, 1, 1, 1}. 0 1 2 3 4 1 5 5 5 5 We note the C 0 continuity in ζ j = 2 5. Each B p i is a piecewise positive polynomial of degree p and has a local support. On each element, at most (p + 1) functions have non-zero values. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 5 / 45

IGA basics Univariate B-Splines Important properties 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Univariate quadratique B-Spline functions defined on the knot vector Ξ = {0, 0, 0, 1 5, 2 5, 2 5, 3 5, 4 5, 1, 1, 1}. 0 1 2 3 4 1 5 5 5 5 We note the C 0 continuity in ζ j = 2 5. Each B p i is a piecewise positive polynomial of degree p and has a local support. On each element, at most (p + 1) functions have non-zero values. The inter-element continuity is defined by the breakpoint multiplicity: the basis is C p m j at ζ j. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 5 / 45

IGA basics Multivariate B-Splines and NURBS For any direction δ, we have p δ, U δ, Ξ δ and n δ. Then we introduce U = (U 1 U 2... U d ), Ξ = (Ξ 1 Ξ 2... Ξ d ) and I = {i = (i 1,..., i d ) : 1 i δ n δ } to define multivariate B-Splines by a tensor product of univariate B-Splines. Given a set of positive weights {ω i, i I}, the Non-Uniform Rational B-Splines functions are defined as rational functions of multivariate B-Spline functions by N p i (ζ) = ω i B p i (ζ) i I ω i B p i (ζ). Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 6 / 45

IGA basics Multivariate B-Splines and NURBS For any direction δ, we have p δ, U δ, Ξ δ and n δ. Then we introduce U = (U 1 U 2... U d ), Ξ = (Ξ 1 Ξ 2... Ξ d ) and I = {i = (i 1,..., i d ) : 1 i δ n δ } to define multivariate B-Splines by a tensor product of univariate B-Splines. Given a set of positive weights {ω i, i I}, the Non-Uniform Rational B-Splines functions are defined as rational functions of multivariate B-Spline functions by N p i (ζ) = ω i B p i (ζ) i I ω i B p i (ζ). S p (Ξ) is the spline space spanned by the functions Ŝ p i (ζ), N p (Ξ) the NURBS space spanned by the functions N p i (ζ). Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 6 / 45

IGA basics Geometrical parametrisation In CAGD, NURBS and splines are widely spread as they are capable of accurately describing various geometries. Given a set of Control Points C i R d, i I, a NURBS parametrization of a curve (d=1), a surface (d = 2) or a solid (d = 3) is given by F : Ω Ω F(ζ) = i I C i Np i (ζ). F Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 7 / 45

IGA basics Geometrical approximation The computational domain Ω is generally decomposed into K non-overlapping subdomains, i.e., K Ω = Ω k, Ω i Ω j =, i j. k=1 Some of the reasons: Geometrical definition purposes Different materials Figure: A tire kindly given by Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 8 / 45

IGA basics Some model problems Strong formulations Static mechanical balance problem div(σ) = f in Ω, u = u D on Γ D, σ n = l on Γ N. Laplace problem u = f in Ω, u = u D on Γ D, u n = l on Γ N. under the following assumptions: linear elasticity ε = 1 2 ( u + ( u)t ) = s u, small displacement-deformation σ = λ Lamé tr(ε) I + 2 µ Lamé ε. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 9 / 45

IGA basics Some model problems Weak formulations Find u V D such that a(u, v) = f (v), v V 0. Laplace problem a(u, v) = u v dx, Ω f (v) = f v dx + l v ds. Ω Γ N Static mechanical balance problem a(u, v) = λ Lamé div(u) div(v) + Ω 2 µ Lamé s u : s v dx, f (v) = f v dx + l v ds. Ω Γ N Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 10 / 45

IGA basics Motivations Each Ω k is parametrised by a NURBS parametrization F k. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 11 / 45

IGA basics Motivations Each Ω k is parametrised by a NURBS parametrization F k. Each subdomain is discretized independently non-conforming meshes at the subdomain interfaces! The approximation space on Ω k is V k,h = {v k = v k F 1 k, v k (N p k (Ξ k )) d P }, i.e., a push-forward of a NURBS space. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 11 / 45

IGA basics Motivations Each Ω k is parametrised by a NURBS parametrization F k. Each subdomain is discretized independently non-conforming meshes at the subdomain interfaces! The approximation space on Ω k is V k,h = {v k = v k F 1 k, v k (N p k (Ξ k )) d P }, i.e., a push-forward of a NURBS space. at the interfaces? Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 11 / 45

Where we are 1 IGA basics 2 Isogeometric mortar methods 3 Applications in contact mechanics Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 12 / 45

Isogeometric mortar methods The discrete mortar problem We define the interface γ k1 k 2 by γ k1 k 2 = Ω k1 Ω k2 k 1, k 2 such that 1 k 1 < k 2 K. For each interface, one of the adjacent subdomains is chosen as the master side and the other one as the slave side. The skeleton Γ is the union of all the interfaces. Ω 1 γ 2 Ω 1 γ1 Ω M(2) Ω S(2) Ω S(1) n Ω 1 3 Ω 2 γ 1 Ω S(1) n 2 Ω M(1) n 1 Ω 2 γ 2 Ω S(2) n 2 Ω M(1) Ω M(2) Ω 3 Figure: Geometrical conforming case (left) and slave conforming case (right) Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 13 / 45

Isogeometric mortar methods The discrete mortar problem Mortar method weak continuity imposition of u at each interface. Discrete mortar saddle point formulation Find (u h, λ h ) V h M h, such that with V h = a(u h, v h ) + b(v h, λ h ) = f (v h ), v h V h, b(u h, φ h ) = 0, φ h M h, K V k,h and b(u h, φ h ) = k=1 Γ u h φ h ds. Choose properly M h Two requirements on each interface to have an optimal method an appropriate approximation order for the multiplier space a uniform inf-sup stability between the primal and multiplier spaces Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 14 / 45

Where we are 1 IGA basics 2 Isogeometric mortar methods The discrete mortar problem Lagrange multiplier spaces A key issue: the evaluation of the mortar integrals 3 Applications in contact mechanics Contact problem definition Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 15 / 45

Isogeometric mortar methods Lagrange multiplier spaces M h = L M l,h where M l,h is the multiplier space on an interface. l=1 From now on, we focus on the one interface case. M h is generated from a parametric multiplier space M h. We consider the following choices for this latter one: choice 1: Mh S p S ( Γ), choice 2: Mh S p S 1 ( Γ), choice 3: Mh S p S 2 ( Γ). Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 16 / 45

Isogeometric mortar methods Lagrange multiplier spaces Choice 1 M h S p S ( Γ) A boundary modification can be necessary. 2 1 B 2 i 0 1 0 h 2h 3h ζ Figure: Left boundary modification of B-Spline functions of degree p = 3 A degree reduction in the boundary elements. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 17 / 45

Isogeometric mortar methods Lagrange multiplier spaces Choice 2 M h S p S 1 ( Γ) Chapelle-Bathe 1 tests to estimate the stability constant 10 0 Stability constant 10 1 10 2 P8 P7 P7 P6 P6 P5 P5 P4 P4 P3 P3 P2 P2 P1 P1 P0 O(E 1 ) 10 3 10 2 10 1 Inverse of the number of elements Unstable pairings 1 The inf-sup test. D. Chapelle, K. J. Bathe. Computers & Structures, Vol. 47 (1993), No. 4/5, pp. 537-545 Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 18 / 45

Isogeometric mortar methods Lagrange multiplier spaces Choice 3 M h S p S 2 ( Γ) Proof for the p S /p S 2 pairing in 2D inf φ S p S 2 sup û S p S 0 Γ φ û dŝ û L 2 φ L 2 α Regularity of the mapping Quasi-uniformity of the meshes Choose û S p S 0 such that 2 xxû = φ D : S p S 0 S p S 1 \ R and D : S p S 1 \ R S p S 2 are bijections Work with Sobolev norms Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 19 / 45

Isogeometric mortar methods Lagrange multiplier spaces inf φ S p S 2 sup û S p S 0 Γ φ û dŝ û L 2 φ L 2 α inf sup φ M h u W h From parametric to physical space Γ φ u ds u L 2 φ L 2 α W h = {u Γ, u V S,h } H 1 0 (Γ) Variable change: Γ φ u ds = Γ φ û ρ dŝ, Super-convergence results a Π : L 2 ( Γ) M h φρ Π( φρ) L 2 (ˆΓ) Ch φ L 2 (ˆΓ) a Superconvergence in Galerkin Finite Element Methods. L. Wahlbin. Lecture Notes in Mathematics (1995), Vol. 1605, Springer, Berlin Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 19 / 45

Isogeometric mortar methods Lagrange multiplier spaces inf φ S p S 2 sup û S p S 0 Γ φ û dŝ û L 2 φ L 2 α inf sup φ M h u W h From parametric to physical space Γ φ u ds u L 2 φ L 2 α W h = {u Γ, u V S,h } H0 1(Γ) Variable change: Γ φ u ds = Γ φ û ρ dŝ, Super-convergence results a For h small enough, the stability holds! a Superconvergence in Galerkin Finite Element Methods. L. Wahlbin. Lecture Notes in Mathematics (1995), Vol. 1605, Springer, Berlin Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 19 / 45

Isogeometric mortar methods Lagrange multiplier spaces Dependence of the stability constant on the degree 10 0 10 0.1 Stability constant 10 0.2 10 0.3 Pp Pp 10 0.4 Pp P(p 2) 0 2 4 6 8 10 12 14 p Exponential dependence in p Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 20 / 45

Isogeometric mortar methods Lagrange multiplier spaces - Summary For V S,h a push-forward of a NURBS space of degree p S, and : M h a push-forward of a spline space of degree p S = problems at cross points. M h a push-forward of a spline space of degree p S 1 = stability is NOK. M h a push-forward of a spline space of degree p S 2 = stability is OK. And more generally, for a push-forward of a spline space of degree p S 2k (k N and k > 1). Pairing p S p S Pairing p S (p S 2) u u h L 2 (Ω) p S + 1 p S + 1 2 λ λ h L 2 (Γ) p S 1 2 p S 1 Table: Optimal order of convergence of isogeometric mortar methods Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 21 / 45

Isogeometric mortar methods A key issue: the evaluation of the mortar integrals We recall the bilinear form b(u, φ) = φ u S ΓS Γ S ds φ u M ΓM Γ S ds. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 23 / 45

Isogeometric mortar methods A key issue: the evaluation of the mortar integrals We recall the bilinear form b(u, φ) = φ u Γ S ds φ u + Γ S ds. A challenge: the evaluation of the second integral due to the product of functions which are defined on different meshes. Finite element litterature Dedicated quadrature rules have been studied. a,b Approximating these integrals taking the quadrature of one side is inducing large (consistency or approximation) errors. a Numerical quadrature and mortar methods. L. Cazabeau, C. Lacour, Y. Maday. Computational Science for the 21st Century, John Wiley and Sons (1997), pp. 119-128 b The influence of quadrature formulas in 2D and 3D mortar element methods. Y. Maday, F. Rapetti, B.I. Wohlmuth. Recent developments in domain decomposition methods. Some papers of the workshop on domain decomposition. ETH Zürich, Switzerland, June 7-8. 2001, pp. 203-221. Springer (2002) Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 23 / 45

Isogeometric mortar methods A key issue: the evaluation of the mortar integrals Slave integration approach One could expect the sensitivity with respect to the slave quadrature to be less than for finite element methods. a(ũ h, v h ) + (v h v + h ) λ h = f (v h ), v h V h, (ũ h ũ+ h ) φ h = 0, φ h M h. Mixed integration approach We also consider another approach which leads to a non-symmetric saddle point problem. It is motivated by different requirements for the integration of the primal and multiplier test functions. a(ũ h, v h ) + v h λ h + v + h λ h = f (v h ), v h V h, (ũ h ũ+ h ) φ h = 0, φ h M h. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 24 / 45

λ Isogeometric mortar methods A key issue: the evaluation of the mortar integrals Numerical results Problem settings Standard Poisson problem u = f Analytical solution slave 2 1 0 1 5 0 2 1 0 y 1 0 x 0.5 1 5 0 0.2 0.4 0.6 0.8 1 x Boundary conditions imposed such that no need to handle a cross point master meshes at refinement level 1 Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 25 / 45

Isogeometric mortar methods A key issue: the evaluation of the mortar integrals Numerical results - Slave integration approach L 2 primal and multiplier errors as a function of the number of additional quadrature points 10 4 u uh L 2 (Ω) 10 5 10 10 0 1 2 3 4 Exact 0 1 2 3 4 5 6 Refinement level number u n λ h L 2 (Γ) 10 6 10 8 10 10 0 1 2 3 4 Exact 10 12 0 1 2 3 4 5 6 Refinement level number P5 P5 Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 26 / 45

Isogeometric mortar methods A key issue: the evaluation of the mortar integrals Numerical results - Mixed integration approach L 2 primal error as a function of the number of additional quadrature points 10 2 10 4 u uh L 2 (Ω) 10 6 10 8 10 10 10 12 0 1 2 Exact O(h 4.5 ) 0 1 2 3 4 5 6 7 Refinement level number P4 P2 Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 27 / 45

Isogeometric mortar methods A key issue: the evaluation of the mortar integrals Numerical results - Mixed integration approach L 2 primal error as a function of the number of additional quadrature points 10 1 10 3 u uh L 2 (Ω) 10 5 10 7 10 9 0 1 2 Exact O(h 2.5 ) 0 1 2 3 4 5 6 7 Refinement level number P4 P0 Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 27 / 45

Where we are 1 IGA basics 2 Isogeometric mortar methods 3 Applications in contact mechanics Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 28 / 45

Applications in contact mechanics Contact problem definition - Strong formulation Assumption: frictionless contact the contact variables are thus the gap g N and the contact pressure σ N defined as: 2 types g N = (x S x M ) n M = [x], σ N = (σ S n S ) n M. Rigid-Deformable contact Deformable-Deformable contact g N = u S n M + (X S X M ) n M g N = (u S u M ) n M +(X S X M ) n M Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 29 / 45

Applications in contact mechanics Contact problem definition - Strong formulation Assumption: frictionless contact the contact variables are thus the gap g N and the contact pressure σ N defined as: 2 types g N = (x S x M ) n M = [x], σ N = (σ S n S ) n M. Rigid-Deformable contact Deformable-Deformable contact On Ω = k Ω k with either k = {S}, or k = {S, M}, the mechanical equilibrium with contact conditions is written div(σ k ) = f k in Ω k, u k = u D,k on Γ D,k, σ k n k = l k on Γ N,k, g N 0, σ N 0, g N σ N = 0 on Γ C,S. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 29 / 45

Applications in contact mechanics Contact problem definition - Weak formulations Different formulations to enforce the contact constraints: penalty, Lagrange multiplier, augmented lagrangian, Nitsche s. In the following, we focus on a Lagrange multiplier formulation. To do so, we introduce a multiplier space M defined by M = {φ H 1/2 (Γ C,S )}, as its subspace M = {φ M, Γ C,S φ [u] ds 0, [u] H 1/2 (Γ C,S ), [u] 0}. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 30 / 45

Applications in contact mechanics Contact problem definition - Weak formulations Different formulations to enforce the contact constraints: penalty, Lagrange multiplier, augmented lagrangian, Nitsche s. In the following, we focus on a Lagrange multiplier formulation. We also introduce the following bilinear and linear forms: b : V M R, b(u, φ) = φ [u] ds, Γ C,S g : M R, g(φ) = φ g ds. Γ C,S Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 30 / 45

Applications in contact mechanics Contact problem definition - Weak formulations Different formulations to enforce the contact constraints: penalty, Lagrange multiplier, augmented lagrangian, Nitsche s. In the following, we focus on a Lagrange multiplier formulation. Contact problem - Lagrange multiplier formulation Find (u, λ) V D M such that { a(u, v) + b(v, λ) = f (v), v V0, b(u, λ φ) g(λ φ), φ M. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 30 / 45

Applications in contact mechanics Contact problem definition - Semi-discrete formulations On each subdomain Ω k with either k = {S} (R-D contact) or k = {S, M} (D-D contact), based on its relative NURBS geometrical parametrization, we introduce the displacement approximation space V k,h = {v k = v k F 1 k, v k (N p k (Ξ k )) du }. We note that in the D-D contact case, the discrete product space V h = V k,h V forms a (H 1 (Ω S Ω M )) du non-conforming k={s, M} space discontinuous over the potential region of contact Γ C,S. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 31 / 45

Applications in contact mechanics Contact problem definition - Semi-discrete formulations On each subdomain Ω k with either k = {S} (R-D contact) or k = {S, M} (D-D contact), based on its relative NURBS geometrical parametrization, we introduce the displacement approximation space V k,h = {v k = v k F 1 k, v k (N p k (Ξ k )) du }. We note that in the D-D contact case, the discrete product space V h = V k,h V forms a (H 1 (Ω S Ω M )) du non-conforming k={s, M} space discontinuous over the potential region of contact Γ C,S. Need to complete the definition of the discrete formulation to define remaining discrete datas. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 31 / 45

Applications in contact mechanics Contact problem definition - To discrete formulations Discrete multiplier space For V S,h a push-forward of a NURBS space of degree p S and : M h a push-forward of a spline space of degree p S = cross points M h a push-forward of a spline space of degree p S 2 = stability OK And more generally, for a push-forward of a spline space of degree p S 2k (k N and k > 1). Discrete gap Use of a lumped L 2 -projection Π into M h defined as Γ Π : R M h, ( Πg N ) j = C,S g N B φ j ds, j = 1,..., n Mh. ds Γ C,S B φ j Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 32 / 45

Applications in contact mechanics Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Numerical resolution strategy The Lagrange multiplier formulation of the contact problem is solved with an active set strategy written as: Find (u h, λ h ) V D,h M h such that a(u h, v h ) + λ h [v h ] ds = f (v h ), v h in V 0,h, ACT ACT φ h [u h ] ds = ACT φ h g ds, φ h in M h. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 34 / 45

Applications in contact mechanics Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Numerical resolution strategy The Lagrange multiplier formulation of the contact problem is solved with an active set strategy written as: Find (u h, λ h ) V D,h M h such that a(u h, v h ) + λ h [v h ] ds = f (v h ), v h in V 0,h, ACT Need to focus on ACT φ h [u h ] ds = ACT the discrete active set region definition, φ h g ds, φ h in M h. suitable integration methods to approximate the mixed term contributions. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 34 / 45

Applications in contact mechanics Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Giving a discrete definition of the dual cone M h, e.g., M h,e = {φ h M h, φ h 0}, M h,w = {φ h M h, φ h B j ds 0, j = 1,..., n Mh }, Γ C,S M h,p = {φ h = α k B k, α k 0, k = 1,..., n Mh }, k consists in giving a discrete definition of the active vs inactive contact region. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 35 / 45

Applications in contact mechanics Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Giving a discrete definition of the dual cone M h, e.g., M h,e = {φ h M h, φ h 0}, M h,w = {φ h M h, φ h B j ds 0, j = 1,..., n Mh }, Γ C,S M h,p = {φ h = α k B k, α k 0, k = 1,..., n Mh }, k consists in giving a discrete definition of the active vs inactive contact region. According to the R-D contact case study, we choose to define it as the support of active multiplier functions. This set of functions is denoted CPλ ACT while the set of inactive ones CPλ INA. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 35 / 45

Applications in contact mechanics Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Active set strategy (i) Initialise CPλ ACT (ii) Compute [ and CP INA λ K B (,CP ACT λ ) B 0 (CPλ ACT, ) (CP ACT λ, CPλ ACT ) ] [ U Λ (CP ACT λ ) ] [ = F G (CP ACT λ ) ] (iii) (iv) Check convergence, i.e.,: CP ACT λ Update CP ACT λ and CP INA λ and CPλ INA stable and go to (ii) until convergence is reached The constraints are checked on the multiplier control values λ j discrete gap control values ( Πg N ) j. The 9 possible cases are: and the λ j < 0, λ j > 0, λ j = 0, ( Πg N ) j = 0 Actif (a) ( Πg N ) j = 0 Inactif (b) ( Πg N ) j = 0 Inactif (c) ( Πg N ) j > 0 Actif (d) ( Πg N ) j > 0 Inactif (e) ( Πg N ) j > 0 Inactif (f) ( Πg N ) j < 0 Actif (g) ( Πg N ) j < 0 Actif (h) ( Πg N ) j < 0 Actif (i) Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 36 / 45

Applications in contact mechanics Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Active set strategy (i) Initialise CPλ ACT (ii) Compute [ and CP INA λ K B (,CP ACT λ ) B 0 (CPλ ACT, ) (CP ACT λ, CPλ ACT ) ] [ U Λ (CP ACT λ ) ] [ = F G (CP ACT λ ) ] (iii) (iv) Check convergence, i.e.,: CP ACT λ Update CP ACT λ [ ] B S B = B M [ ] B B S = B M and CP INA λ and CPλ INA stable and go to (ii) until convergence is reached B S ij = B λ j (x S ) n M (x M ) B S i (x S ) ds, ACT B S ij = B λ i (x S ) n M (x M ) B S j (x S ) ds, ACT B M ij = B λ j (x S ) n M (x M ) B M i (x M ) ds, ACT B M ij = B λ i (x S ) n M (x M ) B M j (x M ) ds. ACT Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 36 / 45

Applications in contact mechanics Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Active set strategy (i) Initialise CPλ ACT (ii) Compute [ and CP INA λ K B (,CP ACT λ ) B 0 (CPλ ACT, ) (CP ACT λ, CPλ ACT ) ] [ U Λ (CP ACT λ ) ] [ = F G (CP ACT λ ) ] (iii) (iv) Check convergence, i.e.,: CP ACT λ Update CP ACT λ and CP INA λ and CPλ INA stable and go to (ii) until convergence is reached B S ij = ACT λ B S j (x S ) n M (x M ) B S i (x S ), B S ij = ACT λ B S i (x S ) n M (x M ) B S j (x S ), B M ij = ACT λ B M j ( x S ) n M (x M ) B M i (x M ), B M ij = ACT λ B S i (x S ) n M (x M ) B M j (x M ). Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 36 / 45

Applications in contact mechanics Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Numerical results - Test 1 - Transmission of a constant pressure Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 37 / 45

Applications in contact mechanics Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Numerical results - Test 1 - Transmission of a constant pressure P2 P2, non-conforming meshes, 4 2 elements for each subdomain Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 37 / 45

Applications in contact mechanics Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Numerical results - Test 1 - Transmission of a constant pressure P2 P2, non-conforming meshes, 64 32 elements for each subdomain Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 37 / 45

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 0.005 0.01 Applications in contact mechanics Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Numerical results - Test 2 conforming meshes non-conforming meshes 0.08 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0.015 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Full contact vs Partial contact at refinement level 2 Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 38 / 45

Applications in contact mechanics Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Numerical results - Test 2 - Full contact L 2 multiplier errors 10 2 λ λh L 2 (ΓC,S) 10 4 10 6 Conf. meshes Mix. int. app. Sla. int. app. O(h 0.5 ), O(h 1 ),O(h 1.5 ) 10 2 10 3 10 4 primal dof number P2 P2 Ref: P4 P4, 511 128 elements for each subdomain Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 39 / 45

Applications in contact mechanics Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Numerical results - Test 2 - Full contact P2 P2 Ref. level nb. 3 Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 40 / 45

Applications in contact mechanics Example of treatment of deformable-deformable contact in a Lagrange multiplier formulation Numerical results - Test 2 - Partial contact L 2 multiplier errors 10 2 λ λh L 2 (ΓC,S) 10 4 10 6 Conf. meshes Mix. Int. App. Slav. int. App. O(h 0.5 ), O(h 1 ),O(h 1.5 ) 10 2 10 4 primal dof number P2 P2 Ref: P4 P4, 511 128 elements for each subdomain Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 41 / 45

Conclusion & perspectives Theoretical and numerical study of isogeometric mortar methods. Study of alternative integration methods to alleviate the construction of the merged mesh. From these former results, proposition of contact methods for rigid-deformable and deformable-deformable contact. Additionally, work on a segmentation process for IGA mortar applications. It has been thought to be suitable for mortar-like contact methods. Perspectives: generalisation of the proposed segmentation process, extension of the methods to the large deformation cases, consideration of contact problems with friction. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 43 / 45

Conclusion & perspectives Isogeometric mortar methods. E. Brivadis, A. Buffa, B. Wohlmuth and L. Wunderlich. Comput. Methods Appl. Mech. Eng. 284 (2015), pp. 292-319. The Influence of Quadrature Errors on Isogeometric Mortar Methods. E. Brivadis, A. Buffa, B. Wohlmuth and L. Wunderlich. Isogeometric Analysis and Applications 2014. Ed. by B. Jttler and B. Simeon. Vol. 107. 2015, pp. 33-50. Isogeometric contact mechanics applications. E. Brivadis, A. Buffa. In preparation. Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 44 / 45

Thanks for your attention Ericka Brivadis Isogeometric mortar & contact January 19th, 2017 45 / 45