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1、Meshless Methods in LS-DYNA An Overview of EFG and SPHMeshless Methods in LS-DYNA: An Overview of EFG and SPHYong Guo Livermore Software Technology CorporationLS-DYNA SeminarStuttgart, Germany November 24, 2021Outline1. Introduction to Meshless Methods 2. EFG and SPH in LS-DYNA 3. EFG Applications 4
2、. SPH Applications 5. Conclusions2LS-DYNA Seminar1. Introduction to Meshless MethodsLS-DYNA Seminar3What is the Meshfree/Meshless/ Particle Method? Physical domain is discretized with particles. Approximation solution is solved at the particles. Shape functions are constructed from the particles; no
3、 mesh required.Meshfree Shape FunctionLS-DYNA Seminar4History and Research TrendMeshfree Method Meshfree Collocation MethodSmooth Particle Hydrodynamics (SPH) Lucy1977, Monaghan 1980, Libersky1993 Finite Point Method Onate et al.1996Meshfree Galerkin MethodElement Free Galerkin (EFG) Belytschko et a
4、l. 1994 Reproducing Kernel Particle Method (RKPM) Liu et al. 1995 Partition of Unity Method Babuska and Melenk 1995 HP-Clouds Duarte and Oden 1996 Free-Mesh Method Yagawa et al. 1996 Natural Element Method Sukumar et al.1998 Meshless Local Petrov-Galerkin Meshfree Method(MLPG) Atluri et al.1998 Loca
5、l Boundary Integral Equation (LBIE) Atluri et al. 1998 Finite Sphere Method Bathe 1998, Particle Finite Element Method Idelsohn et al.2021 Meshfree Least Square Method, (FEM, Control Volume, BEM ) + Meshfree MethodCoupled FEM/Meshfree Method 1995 Extended FEM Method 1999 Finite Particle Method 1999L
6、S-DYNA Seminar5Classification of Particle MethodsImplicit Meshfree Galerkin RKPM, EFG, ContinuumExplicit; Hydrocode Meshfree Collocation SPH, Finite point . Meshfree Galerkin RKPM, EFG, Particle MethodMolecular DynamicsDiscreteOthers: Lattice Boltzmann, Discrete Element, LS-DYNA Seminar6Classificati
7、on of Transient Dynamic CodeLagrangian HydrocodeFEM explicit(LS-DYNA, PAM-CRASH, ABAQUS ) structureSmooth Particle Hydrodynamics (SPH)(LS-DYNA, PAM-CRASH, PRONTO3D )structure, fluid, fluid-structure Mesh-free Galerkin Explicit Method(LS-DYNA, TAHOE, DYNA) structureHydrocodeSemi-Lagrangian (Eulerian)
8、 Hydrocode; AdaptivityMesh-free Galerkin Explicit Method (LS-DYNA) structure, fluid, fluid, fluid fluid-structure, structure, metal forming adaptivity Arbitrary Lagrangian-Eulerian Hydrocode (LS-DYNA, MSC/DYTRAN,ALE3D,CALE ) fluid-structure interaction Eulerian Hydrocode(LS-DYNA, MSC/DYTRAN, ALE3D,
9、DYSMAS ) fluid flowLS-DYNA Seminar7Meshfree Application Range“Meshfree Solution looking for problems”F E.B.C.Material StrengthElastic FluidEquation of StatePSolidFluidGasVelocityMetal Forming Extrusion Incompressible fluid Airbag Forging Particle Finite Foam packing Element-free Element Method Parti
10、cle Airbag Crashworthiness FractureGalerkinSloshingBird strike Explosion PenetrationSplashing HydroplaningSPHMomentumLS-DYNA Seminar 8Problem Looking for Meshfree SolutionMulti-Physics : shear band + history dependent large deformation + failureOut of Lagrangian descriptionNumerical : multi-resoluti
11、on + avoid mesh tangle + failure mechanicsSpectral element method The variationl multiscale method Partition of unity method (strong discontinuity) ALE Eulerian Adaptivity Mesh-free Damage mechanics Cohesive model Discrete element methodLS-DYNA Seminar9Large Deformation SimulationVEFG 5827 nodes EFG
12、 Adaptivity 13661 nodesForceLS-DYNA Seminar 10Overview on Element Free Galerkin Method (EFG)u (x) = wan (x ? xI ) u( xI )?xIh I =1 NPMoving Least -Squares approximation or Reproducing Kernel approximation64444444 4 74444444 4 8 -1 T wan (x ? xI ) = H n ( 0 )M n ( x )H n (x - xI ) wa (x ? xI ) 144444
13、 244444 31 4 24 3n ? th order completeness weighting functionn waI ( xJ ) IJ& + A ?T KA ?1?d = ? A?T R A ?T MA ?1?dwIn a( x ? xI ) = 1, x ? ? ? ? ?Higher-order approximation More neighboring nodes Complicated domain integration Special treatment on B.C. Special treatment in nearly incompressible lim
14、itLS-DYNA Seminar11Overview on Smooth Particle Hydrodynamics (SPH) Basic SPH Equation of MotionStrong FormWeak Formd ?v = ? dt ?x dv 1 ? =? ?x dtmj d i = i vi ? v j Wij , dt j j()v(x) = Tu = wa (x ? s)u(s)ds? i = m jWijjKernel approximation i j dvi = ? m j ( 2 + 2 )Wij , i j dt jmj dvi = ? ( i j )Wi
15、j , dt j i j dEi i = 2 dt i ?v dE =? dt ?x m (v ? v )Wj i j jij ,in LS-DYNA 960, 970, 971LS-DYNA Seminar12Comparison of SPH and EFGSPHExplicit Lagrangian Collocative method Impact/penetration compressible flow 2D and 3D Efficient Difficult Boundary conditionEFGExplicit/implicit Lagrangian/Eulerian G
16、alerkin method Manufacturing Crashworthiness Fracture 2D, 3D and shell Accurate SlowLS-DYNA Seminar13Computational ChallengesAdvantages of Using Meshfree MethodLarge material distortion, e.g., crashworthiness, hyper-velocity impact Moving boundaries, free surface, e.g., fluid and structure interacti
17、on Adaptive procedure,e.g., forging and extrusion Multiple-scale phenomenon, e.g., shear band Moving discontinuities, e.g., crack propagationDisadvantages of Using Meshfree MethodHigh CPU and memory in implicit/explicit analysis (EFG) Complicated in parallel (EFG) Tensile instability and zero-energy
18、 mode (SPH) Difficult essential boundary condition treatment (SPH) Does not pass Patch Test (most mesh-free methods); Dispersed wave properties in coarse modelLS-DYNA Seminar 142. EFG and SPH in LS-DYNALS-DYNA Seminar15Element-Free Galerkin Method in LS-DYNAApplied to solids, shell and fluid (trial
19、version) Fully coupled with finite element model Easy change from finite element formulation to EFG formulation Various formulations for industrial applications More effort spent on improving efficiency Available in SMP and MPP; Explicit/Implicit solverLS-DYNA Seminar16Representative EFG Application
20、s? Rubber industry ? Highly compressible foam ? Defense and safety design ? Solid ? Human dummy and barrier EFG Plane strain #43 ? Adaptive forging simulation EFG Axisymmetric #44 ? Fracture simulation EFG 3D solid #41 #42EFG Basic Features 1. 2. Smoother stress and strain Less sensitive to the disc
21、retization 3. No hourglass control 4. Higher accuracy 5. Natural in adaptivity 6. Higher CPU 7. More memory 8. Difficult in parallel 9. More difficult in theory 10. More developments and refinements on theory? ShellEFG shell #41 EFG shell #42? Metal Forming ? Crashworthiness? FluidEFG 3D fluid #41 (
22、limited version)? Compressible fluid flowLS-DYNA Seminar17Current EFG Formulations for Industrial Applications? Metal materials in Forging/Extrusion analysis: Adaptive formulation ? Foam materials: Semi-Lagrangian kernel formulation Stabilized Method ? Rubber materials: Lagrangian kernel formulation
23、 ? Meshfree Shell: Lagrangian kernel, adaptivity ? Quasibrittle material fracture: Strong discontinuities formulation ? E.O.S. materials: Eulerian kernel formulation (trial version)LS-DYNA Seminar18Current Practice in CrashworthinessCPU time RI FEM: SR FEM : Meshfree (8 I.P.) = 1: 4: 10 Stabilized M
24、eshfree formulation (1 I.P.) + Switch to fully integrated (8 I.P.) RI FEM: SR FEM : Meshfree (8 I.P.) = 1: 4: 35 99% Compression 85% requires Formulation change to Eulerian kernel + data remapping or Smooth meshfree approximationLS-DYNA Seminar19Current Practice in Adaptive EFGCPU time FEM: Meshfree
25、 = 1: 23 Global refinement behaves more robust than local refinement Adaptivity can be controlled by fixed frequency or interactively activated by distortion triggers. Mass scaling is allowable. Element erosion is allowed and surface reconstructed for metal cutting.LS-DYNA Seminar20Smooth Particle H
26、ydrodynamics in LS-DYNAA Lagrangian collocative method explicit Efficient Choices of formulations to improve accuracy Applied for Impact/Penetration, In/compressible Flow Most material laws and all E.O.S are available Coupled with Finite Elements through 3 contacts or hybrid element Implemented in M
27、PP versionLS-DYNA Seminar21SPH Formulations IFORM : Particle approximation theory 0 : standard formulation (default) 1 : renormalized formulation 2 : symmetric formulation 3 : symmetric formulation with renormalization 4 : elliptical formulation 5 : fluid formulation 6 : fluid formulation with renor
28、malizationLS-DYNA Seminar223. EFG ApplicationsLS-DYNA Seminar23Meshfree Applications in Production Level Robustness Efficiency Accuracy ? Meshfree Components in Crashworthiness ModelBarriers; bumpers Car seats Human dummies Crush tubes Windshields Fuel slashing LS-DYNA Seminar24ODB SimulationImpact
29、ForceFEMFEM + high viscosityFEM/Meshfre eLS-DYNA Seminar25Dummy with Side ImpactShell Hyper-elastic JacketSolid Foam RibsCourtesy of GMLS-DYNA Seminar 26Foam Compression SimulationFEMEFGEFG + Semi-Lagrangian KernelFoam materials : Semi-Lagrangian kernelLS-DYNA Seminar 27Rubber Bushing Analysis using
30、 Stabilized EFG MethodMooney-Rivlin Rubber Poissons =0.4995 Stabilized EFG explicit analysis Switched to full integration at t=100 Completion at t=150CPU comparison at t=50Methods CPU S-FEM(#1) 1.0 F-FEM(#2) 4.1 EFG 5.412.9 S-EFG 2.6LS-DYNA Seminar28Crushing TubeFEM Meshfre eLS-DYNA Seminar29Cross J
31、oint ForgingComparisons of Implicit and Explicit AnalysisForceVolume change (4.0%)LS-DYNA Seminar 30Metal ExtrusionExtrusion_model210410 nodes2769 nodesLS-DYNA SeminarEFGAdaptive EFG31Wheel Forging Simulation Interactive Adaptivity*CONTROL_ADAPTIVE 2.5 . *CONTROL_REMESHING_EFG 0.15 0.30 3 1 0.20 3.5
32、 0.80* SMP with 6 CPUs IAT Normalized CPU time # of adaptive stepsTraditional adaptivity 0 1.0 50 3 0.72 22Purely interactive adaptivityLS-DYNA Seminar 32Wheel Forging SimulationInteractive adaptivity triggered by indicators Interactive adaptivity triggered by rate of indicator changeToleranceShear
33、deformation Unbalanced nodal distribution Volumetric changeIndicatort (sec)LS-DYNA Seminar33Wheel Forging SimulationContact forceInternal energyLS-DYNA Seminar34Metal Cutting Simulationw/ interactiveMPPw/o interactiveStop due to local distortion Shell rigid MAT_003Resultant forceLS-DYNA Seminar35Met
34、al Cutting SimulationMaterial damage regionDelete damaged elements starting from surfaceDamaged elements before adaptive re-meshingTool MetalElement erosionX=i ; Xi S ( X) XiiNode X is smoothed to X S : Set of neighboring nodes around X i ( X ) : Local smoothing functionGlobal remeshing and remappin
35、gLS-DYNA SeminarLocal surface smoothing and reconstruction364. SPH ApplicationsLS-DYNA Seminar37High Velocity ImpactCONFIGURATION:Projectile:material: 304 L Steel velocity: 5530 m/s geometry :sphere, = 5 mm Target : material: 6061-T651 Al Thickness : 2.85 mmLS-DYNA Seminar38Bird StrikeLS-DYNA Semina
36、r39Automotive ApplicationsHydro-planeSpin testLS-DYNA Seminar40Hybrid Element Coupling SPH with SolidAdvantage: We have the SPH formulation which can endure quite large deformation and at the same time we have the solid mesh which clearly describes the material interface.Hybrid element: Solid elemen
37、ts constrain SPH nodal locations. SPH elements provide penalty force”against solid nodal motion. Hybrid elements are used as transit layers between SPH elements and solid elements. (shared part ID)Hybrid elements SPH elementsSolid elementsAdvantage: Doesnt need extra tiedinterface between solid and
38、SPHImpact ExampleSet up Final shapeEffective stress Coupling layers5. ConclusionsMeshfree methods can solve problems that finite element methods have difficulties.EFG in LS-DYNA provides engineers a powerful tool with robustness, efficiency and accuracy.EFG has been successfully applied to crashworthiness, metal forging/extrusion, and can be used in metal cutting and fracture analysis.SPH in LS-DYNA is an efficient tool for high velocity impact, penetration and can simulate solid, fluidmaterials.