Appunti lezioni completi FEM

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Fundamental concepts of Finite Elements for Elastic Solids DarioGastaldiComputationalBiomechanicsLaboratory 1.WhatisaFEA?Basicsteps1.Idealization2.Discretization3.Solution2.Typesofproblems3.Examples4.Theelasticproblem.StrongForm5.Theintegralformoftheproblem Content 2 The Finite Element Method (FEM) allows finding an approximate numerical solution to mathematical model of a physical problem What is a FEA? 3 IDEALIZATIONDISCRETIZATIONSOLUTIONPhysicalSystemMathematicalModelDiscreteModelDiscreteSolutionSolution errorSolution + discretization errorSolution + discretization + modeling error Physical model Linear Elasticity•Governing equations•Boundary conditions FE Results What is a FEA? Basic Steps 4 FE Model Mathematical Model Physical ModelIdealization Modelingframework•Governing equations (set of partial differential equations)•Boundary conditions What is a FEA? Basic Steps 5 For each element we calculate: Library of elements Assembly[K] {u} = {F}[Ke] {ue} = {Fe}[Ke] {ue} = {Fe}[Ke] {ue} = {Fe} Analytic solutionF(x,y, z) Numerical solution discretized in nodes What is a FEA? Basic Steps 6 Discretization [K] {u} = {F} Commercial softwares:•Abaqus•Ansys•Adina•Nastran•LS-Dyna ...{u} = [K]-1{F}Solution at nodes ResolutionStiffness MatrixUnknown DisplacementsLoads K PreprocessingPostprocessingWhat is a FEA? Basic Steps 7 Solution Discretization:Divisionin elementsCompute elementalstiffnessmatrices Assembly:Global stiffnessmatrixand Load vectorImpositionofboundaryconditions Resolution of thesystem of equations:Displacements Displacementsin Global orLocalcoordinatesDeterminationof stresses,and deformations* * Numerical process Preprocessing(user) Postprocess(user) •Governing equations•Boundary conditions FEM: Steps in the solution 8 Deformable Solids…