Non-Linear Finite Element Methods in Solid Mechanics Attilio Frangi, attilio.frangi@polimi.it Politecnico di Milano, February 3, 2017, Lesson 1 1
Politecnico di Milano, February 3, 2017, Lesson 1 2 Outline Lesson 1-2: introduction, linear problems in statics Lesson 3: dynamics Lesson 4: locking problems Lesson 5: geometrical non-linearities Lesson 6-7: small strain plasticity Final exam: project? Slides on web-site (search page on www.dica.polimi.it) Send e-mail with photo and name Strong links with the course Non-Linear Solid Mechanics
Planning Basic notions in linear elasticity Politecnico di Milano, February 3, 2017, Lesson 1 3
Planning Time dependent problems Lesson 3: Dynamics (and diffusion) Temperature chart in an engine Impact of a tyre on a surface. Rolling of a tyre on an inclined surface Politecnico di Milano, February 3, 2017, Lesson 1 4
Planning Element Engineering Lesson 4: Pathologies and cures of isoparametric finite elements incompressibility.. rubber (tyres), beams, shells, plasticity.. Politecnico di Milano, February 3, 2017, Lesson 1 5
Planning Non linear quasi-static problems Lesson 5: Introduction. Application to geometrical non-linearities buckling of structures Politecnico di Milano, February 3, 2017, Lesson 1 6
Planning Elastoplasticity Lesson 6: Local issues Lesson 7: Global issues plastic deformation in an exaust-pipe Politecnico di Milano, February 3, 2017, Lesson 1 7
Politecnico di Milano, February 3, 2017, Lesson 1 8 Textbook originated from a course on FEM taught at the Ecole Polytechnique for 5 years 9 modules (1h30 theory + 2h hands on sessions) codes free for download from http://www.ateneonline.it AIM: couple a rigorous theoretical treatment with an the introduction to FEM coding Not only toy codes, but pilot codes prior to serious implementations
Politecnico di Milano, February 3, 2017, Lesson 1 Governing equations in strong form Field equations: If material isotropic: Boundary conditions:
Admissible spaces Space of regular displacements (associated to a bounded energy) Space of fields compatible with boundary data kinematically admissible displacements: statically admissible stresses Space of displacements compatible with zero boundary displacements strain operator! Formulation of the equilibrium problem in linear elasticity (strong form): Politecnico di Milano, February 3, 2017, Lesson 1 10
Weak problem formulation Weak form of local equilibrium equations: T tractions stems from an integration by parts procedure of: corresponds essentially to a form of the Principle of Virtual Power (PPV) Compatibility equation and constitutive law enforced pointwise Politecnico di Milano, February 3, 2017, Lesson 1 11
Variant: eliminate unknown tractions Assume that the two following conditions can be met (we will se later how) restrict w to be kinematically admissible with zero boundary data: choose w in u satisfies a priori boundary conditions in a strong form: u = u D on S u Hence the problem formulation becomes: Politecnico di Milano, February 3, 2017, Lesson 1 12
Galerkin approach for the weak formulation Weak formulation of the linear elastic problem The choice of the unknown and of the virtual fields: leads to the linear system of equations: Politecnico di Milano, February 3, 2017, Lesson 1 13
Galerkin approach for the weak formulation N N leads to the linear system of equations: Politecnico di Milano, February 3, 2017, Lesson 1 14
Galerkin approach: general properties (1/3) Let us express the solution u as u = u N +Δu (Δu is the error of the numerical solution u N with respect to the exact solution u) Virtual field Weak continuum formulation (written for the exact solution u) Weak discrete formulation (written for the approximate solution u N ) The error Δu is orthogonal to every virtual field belonging to the space where the solution is sought (in the sense of the energy norm ) Politecnico di Milano, February 3, 2017, Lesson 1 15
Galerkin approach: general properties (2/3) Deformation energy of with arbitrary kinematically admissible remember u - u N = Δu Property of best approximation: u N is the best approximation of the exact solution u in the selected space of approximation, in the sense of the energy norm: Politecnico di Milano, February 3, 2017, Lesson 1 16
Galerkin approach: general properties (3/3) Deformation energy of exact solution Assumption:. Hence and then: Property 3: if approximates u from below in the energy norm sense: Politecnico di Milano, February 3, 2017, Lesson 1 17
Introduction to FEM Galerkin approach element a partition of Ω into triangular elements sharing nodes This introduces a discretized Ω h Neighbouring elements always share nodes forbidden node notion of conformity two elements can only: - either be well separated - or share one node - or share one edge typically the mesh is created with dedicated codes. In our simple 2D case GMSH Politecnico di Milano, February 3, 2017, Lesson 1 18
Admissible field, step 2: linear interpolation Let us consider one scalar field (e.g. one component of displ. or a temperature field) We draw nodal values of the field v x 2 x 1 nodal values are imposed by boundary conditions The blue line denotes the discretization of an S u region (imposed displacements) Politecnico di Milano, February 3, 2017, Lesson 1 19
Linear interpolation at the local level Let us now focus on a specific element the three nodal values completely define the displacement field within the element v v (k) x 2 x 1 the assumed displacement field is continuous its restriction to each triangle is linear and depends only on nodal values v (l) v(m) x (k) x (l) x (m) Politecnico di Milano, February 3, 2017, Lesson 1 This is only a particular way to express a linear field! 20
Politecnico di Milano, February 3, 2017, Lesson 1 21 Shape functions (local shape functions) N m N k x (m) x (m) N l x (l) x (l) 1 x (m) 1 x(l) x (k) x (k) are called shape functions and are: x (k) sometimes are called local or elemental shape functions as opposed to global shape functions (see later)
Galerkin interpolation Global approximation: specific form of the Galerkin approach with: global shape functions v x 2 Analysis domain: h : kinematically admissible in the sense of FEM approximation x 1 Politecnico di Milano, February 3, 2017, Lesson 1 22
Isoparametric elements: linear triangle revisited x (3) S 1 x x (2) S 2 S 3 physical triangle x (1) linear mapping a 2 2 3 (0,1) (0,0) master triangle conventional ordering of nodes on master element local mapping from master triangle onto physical triangle using area coordinates. Any physical triangle can be mapped on the same master triangle. The mapping is such that the red master nodes are mapped on the blue physical nodes fixed by the user. Linear mapping -> 3 parameters -> 3 nodes 1 (1,0) a 1 Moreover area coordinates coincide with shape functions!!! ISO-parametric (geometry and displacement) Politecnico di Milano, February 3, 2017, Lesson 1 23
Politecnico di Milano, February 3, 2017, Lesson 1 From the parametric space to the physical space 1 2 1 N 1 1 2 N 2 1 3 3 2 3 N 3 1 1 Shape functions are now imagined defined in the parametric space on the master element: N k (a)=a k
Isoparametric elements: quadratic triangle x (2) physical triangle x (5) x (3) x (6) real problem boundary x (4) quadratic mapping x (1) a 2 2 (0,1) (1,0) 3 (0,0) 6 1 a quadratic mapping from master triangle into physical triangle using area coordinates allows to better approximate curved boundaries! 4 (0.5,0.5) 5 (0,0.5) (0.5,0) master triangle a 1 the discretized and real domain still DO NOT coincide everywhere but node position is respected exactly! The same shape functions are then employed to generate the approximation space for displacements Politecnico di Milano, February 3, 2017, Lesson 1 25
Politecnico di Milano, February 3, 2017, Lesson 1 26 Isoparametric elements: quadratic triangle 2 2 2 N 3 5 1 5 5 1 1 N 1 6 3 3 3 4 4 4 N 5 6 6 1 1 1 2 2 2 5 5 5 3 N 4 3 N 6 3 4 N 2 4 4 6 6 6 1 1 1 a 2 2 3 (0,1) 4 5 (0.5,0.5) (0,0.5) (0.5,0) (0,0) 6 1 (1,0) Quadratic shape functions which satisfy the property: a 1 Every shape function is associated to a node and vanishes in all the other nodes
Examples of master and physical elements Politecnico di Milano, February 3, 2017, Lesson 1 27
Examples of master and physical elements Politecnico di Milano, February 3, 2017, Lesson 1 28
Examples of master and physical elements Politecnico di Milano, February 3, 2017, Lesson 1 29
Politecnico di Milano, February 3, 2017, Lesson 1 Shape functions of isoparametric elements: properties conformity convergence
Conformal meshes Two neighbouring elements are required not to overlap, nor to make holes at common boundaries. non conformal non conformal conformal If the families of isoparametric elements described before are employed, this is guaranteed if two elements are: either be well separated or share one node or share one edge, and in this case they have the same number of nodes on the common edge with the same position This is a fundamental benefit of isoparametric elements Politecnico di Milano, February 3, 2017, Lesson 1 31
Example a 2 2 a 2 4 7 3 5 4 8 6 3 6 1 conventional (local) ordering of nodes on T6 master element a 1 global numbering of physical nodes 1 5 2 conventional (local) ordering of nodes on Q8 master element a 1 For each element we select the physical (global) node that will correspond to local node number 1 (this choice is not unique since any corner node will do). The rest of the connectivity flows from this choice connec Important distinction between local and global numbering of nodes. E.g. the 4 th local node in element 1 is the 7 th global node Politecnico di Milano, February 3, 2017, Lesson 1 32
List of degrees of freedom: DOF Politecnico di Milano, February 3, 2017, Lesson 1 33
Galerkin interpolation Global approximation: specific form of the Galerkin approach with: global shape functions v x 2 x 1 : kinematically admissible in the sense of FEM approximation Politecnico di Milano, February 3, 2017, Lesson 1 34
Galerkin interpolation Structure of dof to be memorised!! Field dof associates a number to every nodal component of displacement Specific version fo Galerkin approach with: Politecnico di Milano, February 3, 2017, Lesson 1 35
Global problem formulation Politecnico di Milano, February 3, 2017, Lesson 1
Politecnico di Milano, February 3, 2017, Lesson 1 Global problem formulation element by element computation
Engineering notation Politecnico di Milano, February 3, 2017, Lesson 1 38
Politecnico di Milano, February 3, 2017, Lesson 1 39
Politecnico di Milano, February 3, 2017, Lesson 1 40
Politecnico di Milano, February 3, 2017, Lesson 1 41
Politecnico di Milano, February 3, 2017, Lesson 1 42 Element stiffness matrix At this level no distinction is made between given and unknown displacements (see later)
Politecnico di Milano, February 3, 2017, Lesson 1 43 Numerical integration with Gauss quadrature (a) 1D integrals
Politecnico di Milano, February 3, 2017, Lesson 1 44 Numerical integration with Gauss quadrature (a) 1D integrals (b) 2D or 3D integrals over squares or cubes Cartesian products of 1D formulas
Politecnico di Milano, February 3, 2017, Lesson 1 45 Numerical integration with Gauss quadrature (a) 1D integrals (b) 2D or 3D integrals over squares or cubes (c) 2D or 3D integrals over triangles or tetrahedra Specific formulas not built from 1D formulas Example (triangle, Gauss-Hammer rules with G=3) Integrates exactly every second order polynomial
Politecnico di Milano, February 3, 2017, Lesson 1 46 Numerical integration: bilinear quadrilateral Choice of the order of quadrature The numerical integration is said to be complete if, assuming the Jacobian matrix is constant (non distorted element) the stiffness matrix is integrated exactly.. In our case? G=2 is enough!
Politecnico di Milano, February 3, 2017, Lesson 1 47
Politecnico di Milano, February 3, 2017, Lesson 1 Homeworks for next lesson Revise chapters 1-2-3 of the book In particular: isoparametric elements element integrals and numerical integration assemblage procedure solution convergence operative knowledge of code genlin!! play with examples and exercises provided and create your owns