16.810 16.810Engineering Design and Rapid Prototyping Engineering Design and Rapid PrototypingCAE -Finite Element Method Instructor(s) Prof. Olivier de Weck January 11, 2005Plan for Today  Hand Calculations Aero Æ Structures  FEM Lecture (ca. 45 min)  FEM fundamental concepts, analysis procedure  Errors, Mistakes, and Accuracy  Cosmos Introduction (ca. 30 min)  Given by TA  Follow along step-by-step  Work on CAD and conduct FEA in teams(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 16.810 (16.682) 2Course Concept 16.810 (16.682) 3 todayCourse Flow Diagram (2005) 16.810 (16.682) CAD Introduction Design Optimization CAM Manufacturing Training Hand sketching CAD design Optimization Revise CAD design Assembly Parts Fabrication Problem statement Final Review Test Learning/Review Deliverables (A) Hand Sketch (B) Initial Airfoil (D) Final Design (E) Completed Wing (F) Test Data & (C) Initial Design Structural Tunnel Testing optional (G) CDR Package FEM/Solid Mechanics Xfoil Airfoil Analysis FEM/Xfoil analysis Assembly Cost Estimation Design Intro / Sketch & Wind 4Numerical Method Finite Element Method Boundary Element Method Finite Difference Method Finite Volume Method Meshless Method 16.810 (16.682) 5What 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 (16.682) 6Fundamental Concepts (1) Many engineering phenomena can be expressed by “governing equations” and “boundary conditions” Elastic problems Governing Equation (Differential equation) Thermal problems L() φ+ f = 0 Fluid flow Electrostatics Boundary Conditions etc. B()φ+ g = 0 16.810 (16.682) 7Fundamental Concepts (2) Thermal behavior etc. Governing Equation: () 0L fφ+= Boundary Conditions: () 0B gφ+= []=Ku F A set of simultaneous algebraic equationsFEM Approximate! Geometry is very complex!Elastic deformation {}{} Example: Vertical machining center You know all the equations, but you cannot solve it by hand 16.810 (16.682) 8Fundamental Concepts (3) 1[]{}{} Ku = F {}= [K]− {} u F Property Action Behavior Unknown Property []K Behavior {}u Action {}F Elastic stiffness displacement force Thermal conductivity temperature heat source Fluid viscosity velocity body force Electrostatic Dielectric permittivity electric potential charge 16.810 (16.682) 9Fundamental Concepts (4) It is very difficult to solve 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 Finite element: Small piece of structure 16.810 (16.682) 10Fundamental Concepts (5) Obtain the algebraic equations for each element (this is easy!) Put all the element equations together E E E[KE]{}{} [KE]{}{} [KE]{}{} u = FE u = FE u = FE E E E E E E[KE]{}{} [KE]{}{} [KE]{}{} u = F u = F u = F E E[KE]{}{} [KE]{}{} [KE]{}{} uE = FE uE = FE u = F []{}{} Ku = F 16.810 (16.682) 11Fundamental Concepts (6) Solve the equations, obtaining unknown variabless at nodes. 1{}= [ K]− {} u F[]{}{} Ku = F 16.810 (16.682) 12Concepts - Summary - FEM uses the concept of piecewise polynomial interpolation. - By connecting elements together, the field quantity becomes interpolated over the entire structure in piecewise fashion. - A set of simultaneous algebraic equations at nodes. KxF K x =Ku= F[]{}{} K: Stiffness matrix x: DisplacementF: LoadProperty Action Behavior F 16.810 (16.682) 13Brief History - The term finite element was first coined by Clough in 1960. In the early 1960s, engineers used the method for approximate solutions of problems in stress analysis, fluid flow, heat transfer, and other areas. - The first book on the FEM by Zienkiewicz and Chung was published in 1967. - In the late 1960s and early 1970s, the FEM was applied to a wide variety of engineering problems. - Most commercial FEM software packages originated in the 1970s. (Abaqus, Adina, Ansys, etc.) - Klaus-Jurgen Bathe in ME at MIT Reference [2] 16.810 (16.682) 14Advantages of the FEM Can readily handle very complex geometry: - The heart and power of the FEM Can handle a wide variety of engineering problems - Solid mechanics - Dynamics - Heat problems - Fluids - Electrostatic problems Can handle complex restraints - Indeterminate structures can be solved. Can handle complex loading - Nodal load (point loads) - Element loads - distributed (pressure, thermal, inertial forces) - Time or frequency dependent loading 16.810 (16.682) 15Disadvantages of the FEM A general closed-form solution, which would permit one to examine system response to changes in various parameters, is not produced. The FEM obtains only "approximate" solutions. The FEM has "inherent" errors. Mistakes by users can remain undetected. 16.810 (16.682) 16Typical FEA Procedure by Commercial Software Build a FE modelUser Preprocess Computer Conduct numerical analysisProcess See resultsUser Postprocess 16.810 (16.682) 17Preprocess (1) [1] Select analysis type- Structural Static Analysis - Modal Analysis - Transient Dynamic Analysis -Buckling Analysis - Contact - Steady-state Thermal Analysis - Transient Thermal Analysis Linear Truss2-D[2] Select element typeBeamQuadratic3-D Shell Plate Solid[3] Material properties E,,,, ν ραL 16.810 (16.682) 18Preprocess (2) [4] Make nodes [5] Build elements by assigning connectivity [6] Apply boundary conditionsand loads 16.810 (16.682) 19Process and Postprocess [7] Process- Solve the boundary value problem [8] Postprocess- See the results Displacement Stress Strain Natural frequency Temperature Time history 16.810 (16.682) 20Responsibility of the user 200 mm Fancy, colorful contours can be produced by any model, good or bad!! 1 ms pressure pulse BC: Hinged supports Load: Pressure pulse Unknown: Lateral mid point displacement in the time domain Displacement (mm) Results obtained from ten reputable FEM codes and by users regarded as expert.* Time (ms) * R. D. Cook,