美国麻省理工大学：《Engineering Design and Rapid Prototyping》Finite Element Method

16810(16682) Engineering Design and Rapid prototyping IG a10 Finite Element Method Instructor(s) Prof, olivier de Weck Dr Il Yong Kim January 12, 2004 Massachusetts Institute of Technology

16.810 (16.682) 16.810 (16.682) Engineering Design and Rapid Prototyping Engineering Design and Rapid Prototyping Instructor(s) Finite Element Method January 12, 2004 Prof. Olivier de Weck Dr. Il Yong Kim

A10 Plan for Today FEM Lecture(ca 50 min) FEM fundamental concepts analysis procedure Errors mistakes, and accuracy Cosmos Introduction(ca 30 min) Follow along step-by-step Conduct Fea of your part( ca 90 mi Work in teams of two First conduct an analysis of your CAd design You are free to make modifications to your original model 16.810(16682) Massachusetts Institute of Technology

16.810 (16.682) 2 Plan for Today „ FEM Lecture (ca. 50 min) „ FEM fundamental concepts, analysis procedure „ Errors, Mistakes, and Accuracy „ Cosmos Introduction (ca. 30 min) „ Follow along step-by-step „ Conduct FEA of your part (ca. 90 min) „ Work in teams of two „ First conduct an analysis of your CAD design „ You are free to make modifications to your original model

IG. A10 Course Concept Phase 1 today Problem Sketch by hand CAD CAE Rapid Prototyping/ statement Validation Manufacturin /Test ase 2 Design Optimization Optimum solution Rapid Prototyping/ validatio Manufactural / Test 16.810(16682) Massachusetts Institute of Technology

16.810 (16.682) 3 Course Concept today

1G.810 Course Flow Diagram Learning/Review Problem statement Deliverables Design Intro Hand sketching Design Sketch v1 due today CAD/CAM/CAE Intro CAd design Drawing v1 FEM/Solid Mechanics Overview FEM analysis today Analysis output v wednesday Manufacturing Produce part 1 Part v1 Training Structural Test Test Experiment data v1 "Traini Design Optimization Optimization Design/Analysis output v2 Produce Part 2 Part v2 Test Experiment data v2 Final review 16.810(16682) Massachusets Institute of Technology H

16.810 (16.682) 4 Course Flow Diagram CAD/CAM/CAE Intro FEM/Solid Mechanics Overview Manufacturing Training Structural Test “Training” Design Optimization Hand sketching CAD design FEM analysis Produce Part 1 Test Produce Part 2 Optimization Problem statement Final Review Test Learning/Review Deliverables Design Sketch v1 Analysis output v1 Part v1 Experiment data v1 Design/Analysis output v2 Part v2 Experiment data v2 Drawing v1 Design Intro due today today Wednesday

IG. A10 Numerical Method Finite element method Boundary element Method Finite difference method Finite volume method Meshless method 16.810(16682) Massachusetts Institute of Technology

16.810 (16.682) 5 Numerical Method Finite Element Method Boundary Element Method Finite Difference Method Finite Volume Method Meshless Method

IG. A10 What is the fem? FEM: Method for numerical solution of field problems Description FEM cuts a structure into several elements(pieces of the structure Then reconnects elements at"nodes"as if nodes were pins or drops of glue that hold elements together This process results in a set of simultaneous algebraic equations Number of degrees-of-freedom(DOF Continuum: Infinite FEM: Finite This is the origin of the name, Finite Element Method) 16.810(16682) Massachusetts Institute of Technology

16.810 (16.682) 6 What is the FEM? Description - FEM cuts a structure into several elements (pieces of the structure). - Then reconnects elements at “nodes” as if nodes were pins or drops of glue that hold elements together. - This process results in a set of simultaneous algebraic equations. FEM: Method for numerical solution of field problems. Number of degrees-of-freedom (DOF) Continuum: Infinite FEM: Finite (This is the origin of the name, Finite Element Method)

A10 Fundamental Concepts (1) Many engineering phenomena can be expressed by governing equations"and "boundary conditions Governing equation Elastic problems Differential equation) Thermal problems L()+f=0 Fluid flow Electrostatics Boundary conditions etc B()+g=0 16.810(16682) Massachusetts Institute of Technology

16.810 (16.682) 7 Fundamental Concepts (1) Elastic problems Thermal problems Fluid flow Electrostatics etc. Many engineering phenomena can be expressed by “governing equations” and “boundary conditions” Governing Equation (Differential equation) L f () 0 I  Boundary Conditions B g () 0 I 

A10 Fundamental Concepts(2) Example: Vertical machining center Geometry is Elastic deformation very complex! Thermal behavior etc A set of simultaneous FEM Governing algebraic equations Equation L()+=0 Conditions: B(9)+g=0 Approximate K]{u}={F} You know all the equations, but you cannot solve it by hand 16.810(16682) Massachusetts Institute of Technology

16.810 (16.682) 8 Elastic deformation Thermal behavior etc. Governing Equation: L f () 0 I  Boundary Conditions: B g () 0 I  [ ]{ } { } Ku F A set of simultaneous algebraic equations FEM Approximate! Fundamental Concepts (2) Example: Vertical machining center Geometry is very complex! You know all the equations, but you cannot solve it by hand

A10 Fundamental Concepts (3) IK}={F◆{=[K]{F} Property Action Behavior Unknown Property [ K Behavior u Action F; Elastic stiffness displacement orce Thermal conductivity temperature heat source Fluid vIScosity velocity body force Electrostatic dialectri permittivity electric potential charge 16.810(16682) Massachusetts Institute of Technology

16.810 (16.682) 9 [ ]{ } { } Ku F 1 {} [ ] {}  u KF Property Behavior Action Elastic Thermal Fluid Electrostatic Property [ ] K Behavior { }u Action { }F stiffness displacement force conductivity temperature heat source viscosity velocity body force dialectri permittivity electric potential charge Unknown Fundamental Concepts (3)

A10 Fundamental Concepts (4) It is very difficult to make the algebraic equations for the entire domain Divide the domain into a number of small, simple elements A field quantity is interpolated by a polynomial over an element Adjacent elements share the dof at connecting nodes 昌口昌 Finite element: Small piece of structure 16.810(16682) Massachusetts Institute of Technology 10

16.810 (16.682) 10 It is very difficult to make the algebraic equations for the entire domain Divide the domain into a number of small, simple elements Adjacent elements share the DOF at connecting nodes Fundamental Concepts (4) Finite element: Small piece of structure A field quantity is interpolated by a polynomial over an element