CHAP 3 WEIGHTED RESIDUAL AND ENERGY METHOD FOR 1D PROBLEMS - PowerPoint Presentation

1. CHAP 3 WEIGHTED RESIDUAL AND ENERGY METHOD FOR 1D PROBLEMSFINITE ELEMENT ANALYSIS AND DESIGNNam-Ho Kim

2. INTRODUCTIONDirect stiffness method is limited for simple 1D problemsFEM can be applied to many engineering problems that are governed by a differential equationNeed systematic approaches to generate FE equationsWeighted residual methodEnergy methodOrdinary differential equation (second-order or fourth-order) can be solved using the weighted residual method, in particular using Galerkin methodPrinciple of minimum potential energy can be used to derive finite element equations

3. EXACT VS. APPROXIMATE SOLUTIONExact solutionBoundary value problem: differential equation + boundary conditionsDisplacements in a uniaxial bar subject to a distributed force p(x)Essential BC: The solution value at a point is prescribed (displacement or kinematic BC)Natural BC: The derivative is given at a point (stress BC)Exact solution u(x): twice differential functionIn general, it is difficult to find the exact solution when the domain and/or boundary conditions are complicatedSometimes the solution may not exists even if the problem is well defined

4. EXACT VS. APPROXIMATE SOLUTION cont.Approximate solutionIt satisfies the essential BC, but not natural BCThe approximate solution may not satisfy the DE exactlyResidual:Want to minimize the residual by multiplying with a weight W and integrate over the domainIf it satisfies for any W(x), then R(x) will approaches zero, and the approximate solution will approach the exact solutionDepending on choice of W(x): least square error method, collocation method, Petrov-Galerkin method, and Galerkin methodWeight function

5. GALERKIN METHODApproximate solution is a linear combination of trial functionsAccuracy depends on the choice of trial functionsThe approximate solution must satisfy the essential BCGalerkin methodUse N trial functions for weight functionsTrial function

6. GALERKIN METHOD cont.Galerkin method cont.Integration-by-parts: reduce the order of differentiation in u(x)Apply natural BC and rearrangeSame order of differentiation for both trial function and approx. solutionSubstitute the approximate solution

7. GALERKIN METHOD cont.Galerkin method cont.Write in matrix formCoefficient matrix is symmetric; Kij = KjiN equations with N unknown coefficients

8. EXAMPLE1Differential equation Trial functionsApproximate solution (satisfies the essential BC)Coefficient matrix and RHS vector

9. EXAMPLE1 cont.Matrix equationApproximate solutionApproximate solution is also the exact solution because the linear combination of the trial functions can represent the exact solution

10. EXAMPLE2Differential equation Trial functionsCoefficient matrix is same, force vector:Exact solutionThe trial functions cannot express the exact solution; thus, approximate solution is different from the exact one

11. EXAMPLE2 cont.Approximation is good for u(x), but not good for du/dx

12. HIGHER-ORDER DIFFERENTIAL EQUATIONSFourth-order differential equationBeam bending under pressure loadApproximate solutionWeighted residual equation (Galerkin method)In order to make the order of differentiation same, integration-by-parts must be done twice

13. HIGHER-ORDER DE cont.After integration-by-parts twiceSubstitute approximate solutionDo not substitute the approx. solution in the boundary termsMatrix form

14. EXMAPLEFourth-order DETwo trial functionsCoefficient matrix

15. EXAMPLE cont.RHSApproximate solutionExact solution

16. EXAMPLE cont.

17. FINITE ELEMENT APPROXIMATIONDomain DiscretizationWeighted residual method is still difficult to obtain the trial functions that satisfy the essential BCFEM is to divide the entire domain into a set of simple sub-domains (finite element) and share nodes with adjacent elementsWithin a finite element, the solution is approximated in a simple polynomial formWhen more number of finite elements are used, the approximated piecewise linear solution may converge to the analytical solution

18. FINITE ELEMENT METHOD cont.Types of finite elements 1D 2D 3D Variational equation is imposed on each element. One element

19. TRIAL SOLUTION Solution within an element is approximated using simple polynomials.i-th element is composed of two nodes: xi and xi+1. Since two unknowns are involved, linear polynomial can be used:The unknown coefficients, a0 and a1, will be expressed in terms of nodal solutions u(xi) and u(xi+1).

20. TRIAL SOLUTION cont.Substitute two nodal valuesExpress a0 and a1 in terms of ui and ui+1. Then, the solution is approximated bySolution for i-th element:Ni(x) and Ni+1(x): Shape Function or Interpolation Function

21. TRIAL SOLUTION cont.ObservationsSolution u(x) is interpolated using its nodal values ui and ui+1.Ni(x) = 1 at node xi, and =0 at node xi+1.The solution is approximated by piecewise linear polynomial and its gradient is constant within an element.Stress and strain (derivative) are often averaged at the node.Ni(x)Ni+1(x)xixi+1

22. GALERKIN METHODRelation between interpolation functions and trial functions1D problem with linear interpolationDifference: the interpolation function does not exist in the entire domain, but it exists only in elements connected to the nodeDerivative

23. EXAMPLESolve using two equal-length elementsThree nodes at x = 0, 0.5, 1.0; displ at nodes = u1, u2, u3Approximate solution

24. EXAMPLE cont.Derivatives of interpolation functionsCoefficient matrixRHS

25. EXAMPLE cont.Matrix equationStriking the 1st row and striking the 1st column (BC)Solve for u2 = 0.875, u3 = 1.5Approximate solutionPiecewise linear solutionConsider it as unknown

26. EXAMPLE cont.Solution comparisonApprox. solution has about 8% errorDerivative shows a large discrepancyApprox. derivative is constant as the solution is piecewise linear

27. FORMAL PROCEDUREGalerkin method is still not general enough for computer codeApply Galerkin method to one element (e) at a timeIntroduce a local coordinateApproximate solution within the elementElement e

28. FORMAL PROCEDURE cont.Interpolation propertyDerivative of approx. solutionApply Galerkin method in the element level

29. FORMAL PROCEDURE cont.Change variable from x to x Do not use approximate solution for boundary termsElement-level matrix equation

30. FORMAL PROCEDURE cont.Need to derive the element-level equation for all elementsConsider Elements 1 and 2 (connected at Node 2)AssemblyVanished unknown term

31. FORMAL PROCEDURE cont.Assembly of NE elements (ND = NE + 1)Coefficient matrix [K] is singular; it will become non-singular after applying boundary conditions

32. EXAMPLEUse three equal-length elementsAll elements have the same coefficient matrixChange variable of p(x) = x to p(x):RHS

33. EXAMPLE cont.RHS cont.AssemblyApply boundary conditionsDeleting 1st and 4th rows and columnsElement 1Element 2Element 3

34. EXAMPLE cont.Approximate solutionExact solutionThree element solutions are poorNeed more elements

35. ENERGY METHODPowerful alternative method to obtain FE equationsPrinciple of virtual work for a particlefor a particle in equilibrium the virtual work is identically equal to zeroVirtual work: work done by the (real) external forces through the virtual displacementsVirtual displacement: small arbitrary (imaginary, not real) displacement that is consistent with the kinematic constraints of the particleForce equilibriumVirtual displacements: du, dv, and dwVirtual workIf the virtual work is zero for arbitrary virtual displacements, then the particle is in equilibrium under the applied forces

36. PRINCIPLE OF VIRTUAL WORKDeformable body (uniaxial bar under body force and tip force)Equilibrium equation:PVWIntegrate over the area, axial force P(x) = As(x)xE, A(x)BxFLThis is force equilibrium

37. PVW cont.Integration by partsAt x = 0, u(0) = 0. Thus, du(0) = 0the virtual displacement should be consistent with the displacement constraints of the bodyAt x = L, P(L) = FVirtual strainPVW:

38. PVW cont.in equilibrium, the sum of external and internal virtual work is zero for every virtual displacement field3D PVW has the same form with different expressionsWith distributed forces and concentrated forcesInternal virtual work

39. VARIATION OF A FUNCTIONVirtual displacements in the previous section can be considered as a variation of real displacementsPerturbation of displ u(x) by arbitrary virtual displ du(x)Variation of displacementVariation of a function f(u)The order of variation & differentiation can be interchangeableDisplacement variation

40. PRINCIPLE OF MINIMUM POTENTIAL ENERGYStrain energy density of 1D bodyVariation in the strain energy density by du(x)Variation of strain energy

41. PMPE cont.Potential energy of external forcesForce F is applied at x = L with corresponding virtual displ du(L)Work done by the force = Fdu(L)The potential is reduced by the amount of workWith distributed forces and concentrated forcePVWDefine total potential energy F is constant virtual displacement

42. EXAMPLE: PMPE TO DISCRETE SYSTEMSExpress U and V in terms of displacements, and thendifferential P w.r.t displacementsk(1) = 100 N/mm, k(2) = 200 N/mmk(3) = 150 N/mm, F2 = 1,000 NF3 = 500 NStrain energy of elements (springs)F3321231F3u1u2u3

43. EXAMPLE cont.Strain energy of the systemPotential energy of applied forcesTotal potential energy

44. EXAMPLE cont.Total potential energy is minimized with respect to the DOFsGlobal FE equationsForces in the springs Finite element equations

45. RAYLEIGH-RITZ METHODPMPE is good for discrete system (exact solution)Rayleigh-Ritz method approximates a continuous system as a discrete system with finite number of DOFsApproximate the displacements by a function containing finite number of coefficients Apply PMPE to determine the coefficients that minimizes the total potential energyAssumed displacement (must satisfy the essential BC)Total potential energy in terms of unknown coefficientsPMPE

46. EXAMPLEL = 1m, A = 100mm2, E = 100 GPa, F = 10kN, bx = 10kN/mApproximate solutionStrain energyPotential energy of forcesFbx

47. EXAMPLE cont.PMPEApproximate solutionAxial forceReaction force