Introduce to the finite element method for the numerical solution of partial differential equations, in particular: computational fluid dynamics, transport problems; computational solid mechanics.

Curriculum

Programme

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.

The course will cover the following topics: - Applied Linear Algebra.

- Boundary Value Problems.

- Initial Value Problems

Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

Core Documentation

1) Integral Form at a Glance,note a cura del docente

2) When functions have no value(s): Delta functions and distributions

Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method

Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s

Cap. 2 The weak form of a BVP Cap. 3 The Galerkin method

Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

Type of delivery of the course

Theory and practicals with computers; practicals have a noteworthy role in these lectures C01_Rod1D_traction. Example of Elliptic problem; introduction to the integral form and the differential form of the bilance problems. C02_Heat_Equation. Example of Parabolic problem; Introduction to the non steady problems; notion of flux; anisotropic constitutive relation. Dissipation. Compatibility of sources. Weak constraint. C03_Heat_Equation_Conductive_line. Multi-physics coupling. C04a_Heat_2D_Sphere. Example of domain without boundary: the sphere; definition of tangent derivativesi. C04b_Heat_2D_Torus. Example of domain without boundary: the torus. C05_Laplacian_2D_Ellipsoid_Curvature. Example of domain without boundary: the ellipsoid. Constitutive relation with principale curvatures. C06_L2_Norm. Convergence of parabolic problem. Shape functions and mesh size. C07_Iterative_Solver. Linear solver; iterative solvers; pre-conditioning. C08a_Wave_1D. Introduction to non steady hyperbolic problems. Conservation of energy and elastic waves. C08b_Wave_1D_InitialPulse. 1D wave equation and pulse response. C08c_Wave_2D. 2D Wave equation; anisotropic medium. C09a_Convection_Diffusion_1D. Problema dell'interazione diffusione & convenzione; instabilità delle soluzioni e metodi stabilizzanti C09b_Convection_Diffusion_2D. Convection-diffusion problem. C10_Stabilization. Stabilization of convection-diffusion problem. C11_Elastic_Solid. Linear Solid mechanics. C12_Nonlinear_Elastic_Solid. Non-Linear Solid mechanics. C13_Segregated_Solver. Segregated solvers. C14_Cylinder_Flow. 2D Parabolic problem: fluid mechanics. C15_Navier_Stokes_L_Junction _2D. Example of fluid mechanics. C16_Buoyancy_Free. Transport coupled with fluid mechanics; thermal gradients in fluids. C17_Nematic_Liquid_Crystal. Mathematical modeling of Liquid Crystals. Mathematica M01_Test_Function. Mathematica Notebook for test function. M07_Convection_diffusion. Mathematica Notebook for 1D convection-diffusion problem.Type of evaluation

The students are required to produce a written report containing the description of a select problem and a discussion about the numerical experiments.Mutuazione: 20410421 AN430 - METODO DEGLI ELEMENTI FINITI in Scienze Computazionali LM-40 TERESI LUCIANO

Programme

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.

The course will cover the following topics: - Applied Linear Algebra.

- Boundary Value Problems.

- Initial Value Problems

Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

Core Documentation

1) Integral Form at a Glance,note a cura del docente

2) When functions have no value(s): Delta functions and distributions

Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method

Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s

Cap. 2 The weak form of a BVP Cap. 3 The Galerkin method

Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

Type of delivery of the course

Theory and practicals with computers; practicals have a noteworthy role in these lectures C01_Rod1D_traction. Example of Elliptic problem; introduction to the integral form and the differential form of the bilance problems. C02_Heat_Equation. Example of Parabolic problem; Introduction to the non steady problems; notion of flux; anisotropic constitutive relation. Dissipation. Compatibility of sources. Weak constraint. C03_Heat_Equation_Conductive_line. Multi-physics coupling. C04a_Heat_2D_Sphere. Example of domain without boundary: the sphere; definition of tangent derivativesi. C04b_Heat_2D_Torus. Example of domain without boundary: the torus. C05_Laplacian_2D_Ellipsoid_Curvature. Example of domain without boundary: the ellipsoid. Constitutive relation with principale curvatures. C06_L2_Norm. Convergence of parabolic problem. Shape functions and mesh size. C07_Iterative_Solver. Linear solver; iterative solvers; pre-conditioning. C08a_Wave_1D. Introduction to non steady hyperbolic problems. Conservation of energy and elastic waves. C08b_Wave_1D_InitialPulse. 1D wave equation and pulse response. C08c_Wave_2D. 2D Wave equation; anisotropic medium. C09a_Convection_Diffusion_1D. Problema dell'interazione diffusione & convenzione; instabilità delle soluzioni e metodi stabilizzanti C09b_Convection_Diffusion_2D. Convection-diffusion problem. C10_Stabilization. Stabilization of convection-diffusion problem. C11_Elastic_Solid. Linear Solid mechanics. C12_Nonlinear_Elastic_Solid. Non-Linear Solid mechanics. C13_Segregated_Solver. Segregated solvers. C14_Cylinder_Flow. 2D Parabolic problem: fluid mechanics. C15_Navier_Stokes_L_Junction _2D. Example of fluid mechanics. C16_Buoyancy_Free. Transport coupled with fluid mechanics; thermal gradients in fluids. C17_Nematic_Liquid_Crystal. Mathematical modeling of Liquid Crystals. Mathematica M01_Test_Function. Mathematica Notebook for test function. M07_Convection_diffusion. Mathematica Notebook for 1D convection-diffusion problem.Type of evaluation

The students are required to produce a written report containing the description of a select problem and a discussion about the numerical experiments.Programme

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.

The course will cover the following topics: - Applied Linear Algebra.

- Boundary Value Problems.

- Initial Value Problems

Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

Core Documentation

1) Integral Form at a Glance,note a cura del docente

2) When functions have no value(s): Delta functions and distributions

Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method

Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s

Cap. 2 The weak form of a BVP Cap. 3 The Galerkin method

Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

Type of delivery of the course

Theory and practicals with computers; practicals have a noteworthy role in these lectures C01_Rod1D_traction. Example of Elliptic problem; introduction to the integral form and the differential form of the bilance problems. C02_Heat_Equation. Example of Parabolic problem; Introduction to the non steady problems; notion of flux; anisotropic constitutive relation. Dissipation. Compatibility of sources. Weak constraint. C03_Heat_Equation_Conductive_line. Multi-physics coupling. C04a_Heat_2D_Sphere. Example of domain without boundary: the sphere; definition of tangent derivativesi. C04b_Heat_2D_Torus. Example of domain without boundary: the torus. C05_Laplacian_2D_Ellipsoid_Curvature. Example of domain without boundary: the ellipsoid. Constitutive relation with principale curvatures. C06_L2_Norm. Convergence of parabolic problem. Shape functions and mesh size. C07_Iterative_Solver. Linear solver; iterative solvers; pre-conditioning. C08a_Wave_1D. Introduction to non steady hyperbolic problems. Conservation of energy and elastic waves. C08b_Wave_1D_InitialPulse. 1D wave equation and pulse response. C08c_Wave_2D. 2D Wave equation; anisotropic medium. C09a_Convection_Diffusion_1D. Problema dell'interazione diffusione & convenzione; instabilità delle soluzioni e metodi stabilizzanti C09b_Convection_Diffusion_2D. Convection-diffusion problem. C10_Stabilization. Stabilization of convection-diffusion problem. C11_Elastic_Solid. Linear Solid mechanics. C12_Nonlinear_Elastic_Solid. Non-Linear Solid mechanics. C13_Segregated_Solver. Segregated solvers. C14_Cylinder_Flow. 2D Parabolic problem: fluid mechanics. C15_Navier_Stokes_L_Junction _2D. Example of fluid mechanics. C16_Buoyancy_Free. Transport coupled with fluid mechanics; thermal gradients in fluids. C17_Nematic_Liquid_Crystal. Mathematical modeling of Liquid Crystals. Mathematica M01_Test_Function. Mathematica Notebook for test function. M07_Convection_diffusion. Mathematica Notebook for 1D convection-diffusion problem.Type of evaluation

The students are required to produce a written report containing the description of a select problem and a discussion about the numerical experiments.