FOUNDMENTALS FOR FINITE ELEMENT METHOD CHAPTER1: The Finite Element Method A Practical Course The Finite Element Method A Practical Course

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  • The Finite Element MethodA Practical Course FOUNDMENTALS FOR FINITE ELEMENTMETHOD

    CHAPTER1:

    The Finite Element Method by G. R. Liu and S. S. Quek

    CONTENTSSTRONG AND WEAK FORMS OF GOVERNING EQUATIONSHAMILTONS PRINCIPLEFEM PROCEDUREDomain discretizationDisplacement interpolationFormation of FE equation in local coordinate systemCoordinate transformationAssembly of FE equationsImposition of displacement constraintsSolving the FE equationsSTATIC ANALYSISEIGENVALUE ANALYSISTRANSIENT ANALYSIS (reading materials)REMARKS

    The Finite Element Method by G. R. Liu and S. S. Quek

    STRONG AND WEAK FORMS OF GOVERNING EQUATIONSSystem equations: strong form (PDE), difficult to solve.Weak (integral) form: requires weaker continuity on the dependent variables (e.g., u, v, w).Weak form is often preferred for obtaining an approximated solution. Formulation based on a weak form leads to a set of algebraic system equations FEM.

    The Finite Element Method by G. R. Liu and S. S. Quek

    HAMILTONS PRINCIPLEOf all the admissible time histories of displacement the most accurate solution makes the Lagrangian functional a minimum.

    An admissible displacement must satisfy:The compatibility conditionsThe essential or the kinematic boundary conditionsThe conditions at initial (t1) and final time (t2)

    The Finite Element Method by G. R. Liu and S. S. Quek

    HAMILTONS PRINCIPLEMathematicallywhereL=T-P+Wf(Kinetic energy)(Potential energy)(Work done by external forces)Lagrangian functional

    The Finite Element Method by G. R. Liu and S. S. Quek

    FEM PROCEDUREStep 1: Domain discretizationStep 2: Displacement interpolationStep 3: Formation of FE equation in local coordinatesStep 4: Coordinate transformationStep 5: Assembly of FE equationsStep 6: Imposition of displacement constraintsStep 7: Solving the FE equations

    The Finite Element Method by G. R. Liu and S. S. Quek

    Step 1: Domain discretizationThe solid body is divided into Ne elements with proper connectivity compatibility.All the elements form the entire domain of the problem without any gap or overlapping compatibility.There can be different types of element with different number of nodes.The density of the mesh depends upon the accuracy requirement of the analysis.The mesh is usually not uniform, and a finer mesh is often used in the area where the displacement gradient is larger.

    Triangular elements

    Nodes

    The Finite Element Method by G. R. Liu and S. S. Quek

    Step 2: Displacement interpolationBases on local coordinate system, the displacement within element is interpolated using nodal displacements.nf: Degree of freedoms at a nodend: number of nodes in an element

    fsy

    fsx

    A

    3 (x3, y3)

    (u3, v3)

    2 (x2, y2)

    (u2, v2)

    1 (x1, y1)

    (u1, v1)

    y, v

    x, u

    The Finite Element Method by G. R. Liu and S. S. Quek

    Step 2: Displacement interpolationN is a matrix of shape functionswhereShape function for each displacement component at a node

    The Finite Element Method by G. R. Liu and S. S. Quek

    Displacement interpolationConstructing shape functionsConsider constructing shape function for a single displacement componentApproximate in the formpT(x)={1, x, x2, x3, x4,..., xp} (1D)Basis function

    The Finite Element Method by G. R. Liu and S. S. Quek

    Pascal triangle of monomials: 2D

    xy

    x

    x2

    x3

    x4

    x5

    y

    y2

    y3

    y4

    y5

    x2y

    x3y

    x4y

    x3y2

    xy2

    xy3

    xy4

    x2y3

    x2y2

    10 terms

    Constant terms: 1

    1

    6 terms

    Quadratic terms: 3

    Cubic terms: 4

    Quartic terms: 5

    Quintic terms: 6

    Linear terms: 2

    21 terms

    15 terms

    3 terms

    The Finite Element Method by G. R. Liu and S. S. Quek

    Pascal pyramid of monomials : 3D

    35 terms

    20 terms

    10 terms

    y4

    y3

    y2

    y

    x4

    x3

    x2

    x

    xy

    z

    xz

    yz

    xy2

    x2y

    x2z

    zy2

    z2

    xz2

    yz2

    xyz

    z3

    x3y

    x3z

    x2y2

    x2z2

    x2yz

    xy3

    zy3

    z2y2

    xy2z

    xyz2

    xz3

    z4

    z3y

    Quartic terms: 15

    Cubic terms: 10

    Quadratic terms: 6

    Linear terms: 3

    1

    Constant term: 1

    4 terms

    The Finite Element Method by G. R. Liu and S. S. Quek

    Displacement interpolationEnforce approximation to be equal to the nodal displacements at the nodesdi = pT(xi) i = 1, 2, 3, ,nd orde=P where ,Moment matrix

    The Finite Element Method by G. R. Liu and S. S. Quek

    Displacement interpolationThe coefficients in can be found byTherefore, uh(x) = N( x) de

    The Finite Element Method by G. R. Liu and S. S. Quek

    Displacement interpolationSufficient requirements for FEM shape functions

    1.(Delta function property)2.(Partition of unity property rigid body movement)3.(Linear field reproduction property)

    The Finite Element Method by G. R. Liu and S. S. Quek

    Step 3: Formation of FE equations in local coordinatesSince U= Nde Therefore,e = LU e = L N de= B deStrain matrixorwhere(Stiffness matrix)

    The Finite Element Method by G. R. Liu and S. S. Quek

    Step 3: Formation of FE equations in local coordinatesSince U= Nde orwhere(Mass matrix)

    The Finite Element Method by G. R. Liu and S. S. Quek

    Step 3: Formation of FE equations in local coordinates(Force vector)

    The Finite Element Method by G. R. Liu and S. S. Quek

    Step 3: Formation of FE equations in local coordinatesFE Equation(Hamiltons principle)

    The Finite Element Method by G. R. Liu and S. S. Quek

    Step 4: Coordinate transformation,,where(Local)(Global)

    The Finite Element Method by G. R. Liu and S. S. Quek

    Step 5: Assembly of FE equationsDirect assembly methodAdding up contributions made by elements sharing the node(Static)

    The Finite Element Method by G. R. Liu and S. S. Quek

    Step 6: Impose displacement constraintsNo constraints rigid body movement (meaningless for static analysis)Remove rows and columns corresponding to the degrees of freedom being constrainedK is semi-positive definite

    The Finite Element Method by G. R. Liu and S. S. Quek

    Step 7: Solve the FE equationsSolve the FE equation,

    for the displacement at the nodes, D

    The strain and stress can be retrieved by using e = LU and s = c e with the interpolation, U=Nd

    The Finite Element Method by G. R. Liu and S. S. Quek

    STATIC ANALYSISSolve KD=F for D

    Gauss eliminationLU decompositionEtc.

    The Finite Element Method by G. R. Liu and S. S. Quek

    EIGENVALUE ANALYSIS(Homogeneous equation, F = 0)AssumeLet[ K - li M ] fi = 0(Eigenvector)(Roots of equation are the eigenvalues)

    The Finite Element Method by G. R. Liu and S. S. Quek

    EIGENVALUE ANALYSISMethods of solving eigenvalue equationJacobis methodGivens method and Householders methodThe bisection method (Sturm sequences)Inverse iterationQR methodSubspace iterationLanczos method

    The Finite Element Method by G. R. Liu and S. S. Quek

    TRANSIENT ANALYSISStructure systems are very often subjected to transient excitation. A transient excitation is a highly dynamic time dependent force exerted on the structure, such as earthquake, impact, and shocks. The discrete governing equation system usually requires a different solver from that of eigenvalue analysis. The widely used method is the so-called direct integration method.

    The Finite Element Method by G. R. Liu and S. S. Quek

    TRANSIENT ANALYSIS(reading material)The direct integration method is basically using the finite difference method for time stepping.There are mainly two types of direct integration method; one is implicit and the other is explicit.Implicit method (e.g. Newmarks method) is more efficient for relatively slow phenomenaExplicit method (e.g. central differencing method) is more efficient for very fast phenomena, such as impact and explosion.

    The Finite Element Method by G. R. Liu and S. S. Quek

    REMARKSIn FEM, the displacement field U is expressed by displacements at nodes using shape functions N defined over elements. The strain matrix B is the key in developing the stiffness matrix. To develop FE equations for different types of structure components, all that is needed to do is define the shape function and then establish the strain matrix B. The rest of the procedure is very much the same for all types of elements.

    The Finite Element Method by G. R. Liu and S. S. Quek

    Newmarks method (Implicit)Assume thatSubstitute intoTypicallyg = 0.5b = 0.25

    The Finite Element Method by G. R. Liu and S. S. Quek

    Newmarks method (Implicit)whereTherefore,

    The Finite Element Method by G. R. Liu and S. S. Quek

    Newmarks method (Implicit)Start with D0 and Obtain usingObtain usingObtain Dt and usingMarch forward in time

    The Finite Element Method by G. R. Liu and S. S. Quek

    Central difference method (explicit)(Lumped mass no need to solve matrix equation)

    The Finite Element Method by G. R. Liu and S. S. Quek

    Central difference method (explicit)