機械工程中常用的數(shù)值分析方法-1

1、Finite Element & Numerical Optimization in Mechanical EngineeringByProfs. Ji-Hong ZHU & Tong Gao124H10H10H4HFundamental Finite Element MethodHistory & Basic IdeasThe MatricesTruss & Continuum SolidSolutions & SoftwareOptimization: Non-linear ProgrammingSome ConceptsOne-Dimension SearchGradient Metho

2、dKKT ConditionStructural Optimization & ApplicationsSizing OptimizationShape OptimizationTopology OptimizationSome Applications2FEM= Finite Element MethodFEA= Finite Element Analysis3FEM definition in Math In mathematics, the finite element method (FEM) is a numerical technique for finding approxima

3、te solutions to boundary value problems for partial differential equations. It uses variational methods (the calculus of variations) to minimize an error function and produce a stable solution. 4FEM definition in structural mechanicsThe finite element method (FEM) is a powerful technique originally

4、developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. In the FEM, the structural system is modeled by a set of appropriate finite elements interconnected at points called nodes. Elements may have physical properties suc

5、h as thickness, coefficient of thermal expansion, density, Youngs modulus, shear modulus and Poissons ratio. 5Why FEM?Easy to understand the mechanical behaviors of a simple structure. SpringBarBeam6Why FEM?To understand the mechanical behaviors of complicated structures, Finite Element is Essential

6、. 7Why Optimization?= 1km Range= 5kg on Aircraft= $20k= 16km RangeFor aircraft or aerospace vehicles, weight is a critical determinant of performance, payload capacity, maneuverability and range.National Research Council, US1kg1kg1kg1kgStrive to reduce the weight of each gram!AVICOnce we understand

7、the mechanical behaviors, we can make the designs better. 8More reasons?To Improve Multidisciplinary BehaviorsDynamicElectromagnetismThermal9Episode 1Basic Idea of FEM10Numerical calculation of prACODB11Numerical calculation of prACODB12Numerical calculation of pDiscretization: Mesh into elementsApp

8、roximation: Element solutionEquation: Global solutionrACODBPrecision: Refined mesh13Numerical integration14Numerical integration: DiscretizationABCDEFGHIJ15Numerical integration: Approximation16Numerical integration: Equation17Numerical integration: Precision18Episode 2Basic Formulation of FEM19One

9、SpringLoads, Displacements and the Stiffness20More general, 1 spring, 2 DOFsNode 1Node 2One Spring21One SpringLoad Vector, Displacement vector& the Stiffness Matrix22One SpringIts a Singular Matrix! We need more conditions!23One Spring&FixationHereThe Boundary Conditions (BC)24One SpringElement, Nod

10、e & Stiffness MatrixLoad & BCDisplacementBasic Formulation for One Spring ElementBalance on node Several equationsStiffness MatrixAdd BC Remove equationsSolve the Equations25Two SpringsNode 1Node 2Node 31 Balance on node Several equations26Two Springs2 Stiffness Matrix27Two Springs3 Add BC Remove equations&28Two Springs3 Solve the