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)