Copyright c 2006 Tech Science Press CMES, vol.12, no.3, pp.213-227, 2006 Structured Mesh Refinement in Generalized Interpolation Material Point (GIMP) Method for Simulation of Dynamic Problems Jin Ma, Hongbing Lu, and Ranga Komanduri1 Abstract: The generalized interpolation material point 1 Introduction (GIMP) method, recently developed using a C1 continu- ous weighting function, has solved the numerical noise The material point method (MPM) uses a collection of problem associated with material points just crossing the material points, mathematically represented by Dirac cell borders, so that it is suitable for simulation of rela- delta functions to represent a material continuum (Sul- tively large deformation problems. However, this method sky, Zhou, and Schreyer (1995); Hu and Chen (2003); typically uses a uniform mesh in computation when one Guilkey and Weiss (2003)). A spatially fixed background level of material points is used, thus limiting its effec- grid, and interpolation between grid nodes and mate- tiveness in dealing with structures involving areas of rial points are introduced to track physical variables car- high stress gradients. In this paper, a spatial refinement ried by the material points in the Lagrangian descrip- scheme of the structured grid for GIMP is presented for tion. Field equations are solved on the background grid simulations with highly localized stress gradients. A uni- in the Eulerian description. Physical variables are inter- form structured background grid is used in each refine- polated from the solutions on the background grids to ment zone for interpolation in GIMP for ease of generat- material points back and forth for solution and convec- ing and duplicating structured grid in parallel processing. tion of physical variables. In general, the isoparametric The concept of influence zone for the background node shape functions, same as those used in the finite element and transitional node is introduced for the mesh size tran- method (FEM), are used. As the MPM simulation is in- sition. The grid shape function for the transitional node dependent of the background grid, a structured grid is is modified accordingly, whereas the computation of the usually employed for purposes of simplicity. The move- weighting function in GIMP remains the same. Two ment of the material points represents the deformation of other issues are also addressed to improve the GIMP the continuum. MPM has been demonstrated to be capa- method. The displacement boundary conditions are in- ble of handling large deformations in a natural way (Sul- troduced into the discretization of the momentum conser- sky, Zhou, and Schreyer (1995)). However, primarily due vation equation in GIMP, and a method is implemented to to the discontinuity of the gradient of the interpolation track the deformation of the material particles by tracking function at the borders of the neighboring cells, artificial the position of the particle corners to resolve the prob- noise can be introduced when the material points move lem of artificial separation of material particles in GIMP just across the grid cell boundaries, leading to simulation simulations. Numerical simulations of several problems, instability for MPM. The generalized interpolation ma- such as tension, indentation, stress concentration and terial point (GIMP) method, introduced by Bardenhagen stress distribution near a crack (mode I crack problem) and Kober (2004) can resolve this problem. In GIMP are presented to verify this refinement scheme. aC1 continuous interpolation function is used and each material point/particle occupies a region. GIMP has been keyword: GIMP, Material Point Method, Mesh refine- demonstrated to be stable and capable of handling rela- ment,FEM,MPM tively large deformations (Ma, Lu, Wang, Roy, Hornung, Wissink, and Komanduri (2005)). The current MPM typically uses a uniform background mesh for solving the field equations. However, this is 1 Correspondence author, Tel: 405-744-5900; Fax: 405-744-7873; not efficient when stress gradients are high such as stress e-mail: [email protected]. All authors are with the School of Mechanical and Aerospace Engineering, Oklahoma State Univer- concentrations in a plate with a hole, or the stress field of sity, Stillwater, OK 74078 a workpiece under indentation. In contrast, transitional 214 Copyright c 2006 Tech Science Press CMES, vol.12, no.3, pp.213-227, 2006 mesh is effective in solving problems involving rapidly 2GIMP varying stress in an area. Wang, Karuppiah, Lu, Roy, For the purpose of completeness, the basic equations in and Komanduri (2005) have presented a method using GIMP (Bardenhagen and Kober (2004)) are summarized an irregular background mesh to deal with problems in- here. In dynamic simulations, the mass and momentum volving rapidly varying stress, such as stress field near a conservation equations are given by crack. However, this approach does not use regular struc- tured background mesh so that mesh generation encoun- dρ +ρ∇· v = 0, and (1) ters the same difficulty as FEM, and leads to the loss of dt some advantage of MPM on the ease of generating mesh for a complex problem. ρa = ∇σσ +b in Ω, (2) The use of structured grid in GIMP has facilitated the implementation of GIMP in parallel processing. A re- where ρ is the material density, a is the acceleration, σ finement scheme based on splitting and merging material and b are the Cauchy stress and body force density, re- particles was proposed by Tan and Nairn (2002). Re- spectively. The displacement and traction boundary con- cently, a multilevel refinement algorithm has been devel- ditions are given as oped for parallel processing using the structured adap- tive mesh refinement application infrastructure (SAM- u = u on ∂Ωu, (3) RAI) (Hornung and Kohn (2002); Ma, Lu, Wang, Roy, Hornung, Wissink and Komanduri (2005)). The compu- tational domain was divided into multiple nested levels τ = τ on ∂Ωτ, (4) of refinement. Each grid level is uniform but has a differ- ent cell size. Smaller material particles and smaller cell where ∂Ωu ⊂ ∂Ω,∂Ωτ ⊂ ∂Ω and ∂Ωu ∩∂Ωτ = 0. In vari- sizes are used in each finer level. Two neighboring lev- ational form, the momentum conservation equation can els are connected by overlapped material particles of the be written as same size and data communication between levels is per- Z formed at predefined intervals. However, the refinement ρa · δvdx through material particles requires extra communication Ω Z Z Z and simulation time. In this paper, a refinement for GIMP = ∇σσ · δvdx + b· δvdx −α (u−u) · δvdx, (5) based on the transitional grid nodes is developed. This Ω Ω ∂Ω refinement is natural and does not involve extra simula- u tion time. Moreover, the refined grid remains uniformly where δv is an admissible velocity field, α is a penalty structured in each refinement region. parameter we introduce herein to impose the essential While the problem associated with artificial noise has boundary conditions and α >> 1, Atluri and Zhu (1998), been resolved with the use of GIMP method, it has been Atluri and Zhu (2000). Applying the chain rule, ∇ · σ · observed recently that material separation could occur if δv = ∇ · (σ · δv) −σ : ∇δv, and the divergence theorem, the deformation of the material particles was not tracked, Eq. (5) can be written as Guilkey (2005). Tracking the deformation of material Z Z particles properly in GIMP is necessary especially when ρa · δvdx + σ : ∇δvdx the material particles are stretched. In this paper, an ap- Ω Ω proach is developed for tracking the particle deformation Z Z to resolve material point separation problem. This pa- = b· δvdx + τ · δvdS per focuses on the refinement scheme for structured grid. Ω ∂Ωτ Z Z Several numerical problems, such as tension, indenta- + τ · δvdS−α (u−u) · δvdS (6) tion, stress concentration and stress distribution near a u ∂Ω ∂Ω crack (mode I crack problem) were simulated to verify u u this refinement algorithm, as well as to demonstrate the where τ is the resultant traction due to the displacement effectiveness of tracking particle deformations. u boundary condition on ∂Ωu. In GIMP, the domain Ω is Structured Mesh Refinement 215 discretized into a collection of material particles, with Ωp given as as the domain of particle p. The physical quantities, such Z 1 as the mass, stress and momentum can be defined for Sip = χp(x)Si(x)dx. (10) Vp Ω∩Ω each particle. For example, theR momentum for particle p p can be expressed as pp = ρ(x)v(x)χp(x)dx,where Ω p The weighting function in GIMP is C1 continuous and ( ) χ ( ) v x is the velocity and p x is the particle characteris- satisfies partition of unity. The momentum conservation, tic function. The momentum conservation equation can Eq. (8), can be solved at each node to update the nodal be discretized as momentum, acceleration, and velocity. These updated Z Z χ nodal quantities can be interpolated to the material par- p˙ p p · δ + σ χ δ ∑ vdx ∑ p p : vdx ticles to update the particles, as given by Bardenhagen Vp p Ω∩Ω p Ω∩Ω p Z p Z and Kober (2004). It may be noted that the mass of each m χ = ∑ p p b· δvdx +∑ τ · δvdx material particle does not change, so that the mass con- p Vp p servation equation is satisfied automatically. Ω∩Ωp ∂Ωτ∩Ωp Z Z In the discretization of the weak form of the momentum +∑ τu · δvdS−α∑ (u−u) · δvdS (7) conservation equation, a background grid is used. How- p p ∂Ωu∩Ωp ∂Ωu∩Ωp ever, the computation is independent of the grid from one R increment to another. Hence a spatially fixed structured where Vp = χp(x)dx is the particle volume. Intro- grid can be used for convenience. In the background grid, Ω∩Ω p no nodal connectivity is required and the integration is ducing a background grid and the grid shape function never performed on the element domain.