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# Class Note for CE 264 with Professor Marzougui at GW (3)

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Date Created: 02/07/15

Implicit Analysis Lecture 5 CE 264 Nonlinear Finite Element Modeling and Simulation Explicit vs Implicit o Explicit A direct computation of the dependent variables can be made in terms of known quantities Unknown appears only on one side of the equation 39 Xt7 Xt 17 7W0 0 Implicit The dependent variables are defined by coupled sets of equations and either a matrix or iterative technique is needed to obtain the solution Unknown appears on both sides of the equation xtl7 xt h rum7 c5254 Lemues Slide 12 Explicit Ls DYNA o Internal and external forces are summed at each node point and a nodal acceleration is computed by dividing by nodal mass M XL Fatima L Emma L o Solution is advanced by numerical integration of the above computed acceleration in time o Courant condition limits largest stable time step o Typically requires many relatively inexpensive time steps o Well suited for dynamic simulations such as impact and crash short duration c5254 Lemues Slide 13 Implicit Ls DYNA o A global stiffness matrix is computed inverted and applied to the nodal out of balance force to obtain a displacement increment Mxm 1 Kkm l Fumlw l mellw MIMW o Large numerical effort required to form store and factorize the stiffness matrix o Typically involve a relatively small number of expensive time steps o Well suited for static and quasistatic simulations 9 Quasi static analysis time represents a monotonically increasing parameter which characterizes the evolution of the loading c5254 Lemues Slide 14 Activating Implicit Analysis o Three types of analysis can be performed 9 Fully explicit default 9 Fully implicit 9 Explicit followed by implicit switching o Keywords 9 CONTROLIMPLCTGENERAL 9 CONTROLIMPLCITSOLVER 9 CONTROLIMPLICITSOLUTON 9 CONTROLIMPLCITAUTO 9 CONTROLIMPLCITSTABLIZATION 9 CONTROLIMPLCITDYNAMCS 9 CONTROLIMPLCITEGENVALUE Not all features are available in implicit modequot E5254Letures Slide 15 Implicit Keywords CONTROLIMPLICITGENERAL required 9 Actlvates lmpllclt mode 9 Deflrles lmpllclttlme step slze CONTROLIM PLICITSOLVER op ional 9 Performs tne cpu mtenslve stlffrless matrlx lrlverslorl movqu 9 Parameters forllrlearequatlorl solver 9 Does NOT lrlvoke a lmearanalvsls CONTROLMPLCTSOLUTIONopional 9 Parameters forllrlearorrlorlllrlearequatlorl solverNeWtorlrbased metnods 9 Controls lteratlve equlllbflum Search convergence 9 llrlear analvsls selected nere a speclal case Wnere no lteratlons are performed CONTROLIMPLICITAUTOoptional 9 Actlvates automatlctlme step control 9 Default ls xed tlme step slze results m error termmatlon lf anv steps fall to converge c5254 Lemme 5 Slide 15 Implicit Numerical Methods o Implicit governing equations contain two problems to solve o Nonlinear Problem 9 Find displacements X which satisfy equilibrium FEgt Fmt 9 Both K FM and FW can be nonlinear functions ofx 9 Iterative search employed using Newtonbased methods o Linear Problem 9 Solve system oflinear algebraic equations kxf 9 Must solve during every nonlinear iteration 9 Great CPU and memory cost makes this problem important CE 254 Ledures Slide 17 Linear Equation Solver o A linear system of equations ofthe form K uR must be solved within each equilibrium iteration o Stiffness matrix K is inverted and applied to the residual load R yielding a displacement increment u o Storing and solving this linear system represent a large portion of the memory and CPU costs of an implicit analysis o Several different linear equations solvers are available including direct Gaussian elimination and iterative conjugate gradient Lanczos methods o Very important to allow enough memory forthe stiffness matrix factorization E5254Lemres Slide is Nonlinear Equation Solver 9 Several different nonlinear equations solvers are available All of them are iterative 9 When the norms of displacement and energy are reduced below user prescribed tolerances default 10e 3 and 10e2 respectively equilibrium is reached within sufficient accuracy the iteration process is said to have converged and the solution proceeds to the next time step r5254 Lemues Slide 19 Element Formulations 0 Selected using SECTION keywords 0 Default elements often use single point integration with hourglass control 9 Good for explicit where element costs can dominate solution 9 Bad for implicit since hourglass modes cause convergence trouble 0 For implicit analysis it is generally more effective to use more expensive element formulations which are less susceptible to hourglass instability r5254 Lemues Slide nu Hourglass o Crash codes use a 1point integration scheme for elements to achieve fast computation o This causes rank de ciency in stiffness matrix and causes hourglass modes or zero energy modes o Two main hourglass modes IIIPlane Mode 9 In plane membrane mode 9 Out ofplane W mode Outofplane Mode r5254 Lemues Slide 111 Hourglass Control o Tips for Hourglass control 9 Lower Young s modulus than steel hourglass may occur Increase hourglass control coef cient 9 Connecting a 1D element to a shell may generate large hourglassing Using triangular elements may help 9 Improving connections and re ning mesh might help reduce or eliminate hourglass Good l Good Bad Bad E5254Lecmres Slide 112 Hourglass Control Bad Bad Good E5254Lecmres Slide 113 Element Formulations for Implicit Analysis o Shell Elements 9 SIR Hughes Liu shell element 6 2x2 selectivereduced integration 6 DOF per node dx dy dz rx ry rz Most expensive 9 Fullyintegrated shell element16 2x2 integration with enhanced strain formulation 6 DOF per node dx dy dz rx ry rz Least expensive of 2x2 elements o Solid Elements 9 Hughes Liu brick element2 3 DOF per node dx dy dz 2x2x2 selective reduced integration Most expensive r5254 Lemues Slide 114 Element Formulations for Implicit Analysis 0 Beam Element 9 HughesLiu 1 6 DOF per node dx dy dz rx ry rz One integration point along length 0 Spring Element 9 2 node 3 DOF per node dx cly dz orrx ry rz 9 Springs act only in one direction no shear strength unlike beam elements 9 Nonlinear FD behavior and failure can cause convergence trouble E5254Lecmres Slide 115 Implicit Contact Interfaces 0 Several contact interfaces are available for implicit analysis 9 CONTACTSURFACETOSURFACE 9 CONTACTNODESTOSURFACE 9 CONTACTAUTOMATCSINGLESURFACE etc o All implicit contact interfaces use the penalty method 0 Oriented normal vectors are recommended 0 Automatic contact types often fail for implicit r5254 Lemues Slide us Loads o Ramp up load from zero to final value 0 Load curve should extend beyond the termination time for stability Force Tim e r5254 Lemues Slide 117 Automatic 77me Step Control 9 CONTROLIMPLICITAUTO Automatic time step control adjusts stepsize during the simulation gt Very persistent reliable Step size varied such that solving for equilibrium in each step is equally difficult 0 After successful steps Compare iteration count to target value ITEOPT Increasedecrease size of next step if difference exceeds ITEWIN o Afterfailed steps Decrease step size Back up repeat failed step with new DT 9 E5254Lecmres Slide us Implicit Analysis Examples CE 264 Nonlinear Finite Element Modeling and Simulation Model Setup o Honeycomb Block 1225 x1225x 150 CS Area 12252 Length 150 mm o Implicit Analysis Crosshead speed 10 mmsec Output waorc and nodout E5254Lecmres Slide 127 Dyna Input Dyna Input suess mum sum Vs Humm vuwm a n M as no n us as 1 n 5 n s Mame Volume Dyna Output Nnva Fame Noam mun gtlt mpmmem Wm mummy my 254 Lectures Slide 121 Force vs Displacement Force vs Displacement crossp ot onnnn x m n 2n 4n an m m m Dgtsplauemenltnvm1 254 Lectures Slide 123 254 Lectures Slide 3922 Stress vs Relative Volume 5 Xixisy 9 RVo Current volInitial vol 9 RVo L1 L2L3 IL1L2L3 9 RVo L3 IL3 9 RVo150l150 o Yaxis 9 Stress ForceArea 9 Stress Fl12252 E5254Lemues 51119 124 Comparison um mum om Oulpm quotn39 n v a aquot n n as 77 En n mmvme n v a aquot n 77 En n H mm anwue E5254Lecmres 51m 125

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