Finite Element Methods & Model Reductions by J. Monnier ([email protected]) INSA Toulouse, Applied Mathematics department. March 2021 * Variational forms - weak solutions - mathematical analysis Finite Element methods Reduced models Models with weak constraints * Full content of the course with exercises and programming practicals: please consult the INSA Moodle page. 1 (L) A coarse FE mesh in Earth Sciences. (R) FE analysis of an engine: Von Mises stress values. Images source: Contents 1 Analysis of Elliptic Problems: Variational Forms, Weak Solutions 5 1.1 Introduction . 5 1.2 From the classical formulation to the weak formulation . 6 1.2.1 Domain regularity and basic recalls . 6 1.2.2 Weak formulation in the classical spaces Ck(Ω)¯ .............................. 8 1.2.3 Recalls of functional analysis . 9 1.2.4 Weak formulation in the Sobolev spaces and Lax-Milgram’s theory . 11 1.3 The Laplace-Poisson equation: mathematical analysis . 14 1.3.1 The Dirichlet boundary conditions case . 14 1.3.1.1 The model . 14 1.3.1.2 From the classical to the variational form . 14 1.3.1.3 Well-posedness of the model . 15 1.3.1.4 Equivalence of the variational form with the original equation . 15 1.3.1.5 Symmetric case: equivalence with the minimun of energy . 16 1.3.1.6 The non homogeneous Dirichlet condition case . 17 1.3.2 The Neumann boundary conditions case . 18 1.3.2.1 The weak formulation . 18 1.3.2.2 Well posedness (existence - uniqueness) . 18 1.3.2.3 Equivalence with the equations of the BVP . 18 1.3.2.4 Energy estimation & stability inequality . 19 1.3.3 On the regularity of the solution . 20 1.3.3.1 Regular data - regular solution . 20 1.3.3.2 Typical singularity origins . 20 1.3.4 The transmission boundary condition . 22 2 Finite Element Methods 23 2.1 Fundamentals . 24 2.1.1 Internal approximation & discrete weak formulation . 24 2.1.2 On FE meshes . 26 2.1.3 The (linear) algebraic system . 28 2.1.4 A-priori error estimation . 29 2.1.5 Building up a good FE space Vh ...................................... 29 2.1.5.1 On the Galerkin method . 29 2.1.5.2 Required features of any FE space Vh .............................. 30 2.2 The Pk-Lagrange FE . 31 2.2.1 The P1-Lagrange FE in 1D . 31 2.2.2 The Pk-Lagrange FE in nD......................................... 34 2.2.2.1 Triangulation of Ω ......................................... 34 2.2.2.2 The FE space Vh & basis functions . 34 2.2.2.3 The classical higher order Pk-Lagrange FE (k = 2; 3) ..................... 35 2.3 FE code kernel: the assembly algorithm . 36 2.3.1 The assembly algorithm & elementary matrices . 36 2.3.1.1 The linear system coefficients to be computed . 36 2.3.1.2 The assembly algorithm . 38 2.3.1.3 Data structures required from the mesh (resumed) . 38 2 CONTENTS 3 2.3.2 How to introduce the Dirichlet boundary conditions ? . 40 2.3.3 Change of variables onto the reference element K^ ............................. 41 2.3.3.1 The geometric change of variable onto K^ ............................ 41 2.3.3.2 Isoparametric FE . 42 2.3.4 On triangles & tetrahedra (n-simplexes): barycentric coordinates . 43 2.3.4.1 The barycentric coordinates . 43 2.3.4.2 Lattices . 43 2.4 Convergence and error estimation . 44 2.4.1 Interpolation operator & error . 44 2.4.2 FE error estimation in the energy space V ................................. 45 2.4.2.1 The general a-priori error estimation . 45 2.4.2.2 Typical cases . 45 2.4.2.3 On the numerical integration errors . 46 2.4.3 Measuring the convergence order: code validation . 46 2.4.4 On non optimal FE scheme order: presence of singularity . 48 2.4.5 Error estimation in norm L2(Ω) ...................................... 48 2.5 Hermite FE . 50 2.6 Non-linear cases: linearization . 51 2.7 Advective term stabilization . 53 2.7.1 Advective(-diffusive) equations . 53 2.7.2 Standard FE scheme = centered Finite Difference scheme . 54 2.7.3 1D case: explicit solutions - instabilities . 55 2.7.4 Stabilization techniques: SD, SUPG, GLS . 57 3 Projection-Based Reduced Models 60 3.1 Fundamentals . 60 3.1.1 Basic principles . 61 M N 3.1.2 Solutions manifolds & the Kolmogorov -width . 62 3.2 The POD-Galerkin based reduction method . 63 3.2.1 Construction of the Reduced Basis . 63 3.2.1.1 Definition of the snapshot space VM ............................... 63 3.2.1.2 Construction of the reduced space VP OD ............................ 63 3.2.1.3 Relationship between POD and Singular Value Decomposition (SVD) . 64 3.2.1.4 Change of basis: from the snapshot space VM to the FE space Vh . 65 3.2.2 An error estimation . 66 3.2.3 The POD algorithm . 67 3.2.4 Advantages & disadvantages of the POD-based reduction method . 67 3.2.4.1 Summary of the method . 67 3.2.4.2 Advantages & disadvantages of the method . 67 3.2.4.3 Non linear systems: interpolation may be made by an Artificial Neural Network (ANN) . 69 3.2.4.4 An alternative method: the greedy algorithm . 69 3.3 Model reduction in Python & numerical example(s) . 69 3.3.1 Python libraries . 69 3.3.2 Numerical example(s) . 70 4 Weak constraint(s): mixed formulations 72 4.1 The (Navier-)Stokes fluid flow model . 72 4.1.1 The flow model(s) . 72 4.1.2 Formulation in the divergence free space Vdiv ............................... 73 4.1.3 Formulation in variables (u; p): a mixed formulation . 73 4.1.4 The incompressibility constraint: p is the Lagrangian multiplier . 74 4.1.5 Discrete form & linear system . 75 4.1.6 On the Ladyzhenskaya–Babuška–Brezzi (LBB) inf-sup condition . 77 4.2 Mixed formulations: other examples . 78 4.2.1 General form & origins . 78 4.2.2 Dirichlet boundary condition . 79 4.2.3 Non-penetration boundary condition . ..