Meshless methods offer interesting properties for the simulation of bulk forming\ud
processes. This research concerns the investigation of the stabilized conforming nodal integration scheme (SCNI) for use in metal-forming processes. Two tests are carried out. Firstly, the performance of SCNI is compared to a standard integration scheme. The performance seems problem specific. Secondly the footing of a piece of nearly incompressible material is used for testing the locking behavior of the method. No volumetric locking was found

Boogaard, A.H. van den

Quak, W.

English

Meshless methods offer interesting properties for the simulation of bulk forming processes. This research concerns the investigation of the stabilized conforming nodal integration scheme (SCNI) for use in metal-forming processes. Two tests are carried out. Firstly, the performance of SCNI is compared to a standard integration scheme. The performance seems problem specific. Secondly the footing of a piece of nearly incompressible material is used for testing the locking behavior of the method. No volumetric locking was found

van den Boogaard, Antonius H.

University of Twente Research Information

International Conference on Particle-Based MethodsParticles 2009E. Oñate and D.R.J. Owen (Eds)c©CIMNE, Barcelona, 2009NODAL INTEGRATION OF MESHLESS METHODSW. Quak∗, A.H. van den Boogaard∗∗Department of Mechanical EngineeringUniversity of TwenteP.O. Box 217, 7500 AE Enschede, The Netherlandse-mail: w.quak@ctw.utwente.nl, web page: http://www.tm.ctw utwente.nlKey words: nodal integration, SCNI, meshless methods, formingAbstract. Meshless methods offer interesting properties for the simulation of bulk formingprocesses. This research concerns the investigation of thestabilized conforming nodal integra-tion scheme (SCNI) for use in metal-forming processes. Two tests are carried out. Firstly, theperformance of SCNI is compared to a standard integration scheme. The performance seemsproblem specific. Secondly the footing of a piece of nearly incompressible material is used fortesting the locking behavior of the method. No volumetric locking was found.1 INTRODUCTIONFinite element simulations of large-deformation bulk forming processes in a Lagrangian for-mulation can be problematic. When simulating for instance an extrusion or forging process,many re-meshing steps need to be taken to avoid too much mesh distortion. After a re-meshingstep, the material data and the state variables have to be mapped from the old mesh to the newmesh. This mapping can cause inaccuracies.It is expected that meshless methods avoid these mesh-related problems since their shapefunctions are not defined on a mesh and their nodal connectivity is constantly re-evaluatedduring the simulation.Secondly, the particle character of meshless methods mightcontribute to avoid the mappingof state variables. If material points and nodes coincide, th convection of state variables be-comes superfluous. Especially for history dependent material models this will be a beneficialproperty. However, having material points and nodes a the same position will require a nodalintegration scheme. This type of integration is usually unstable. Therefore the stabilized con-forming nodal integration scheme (SCNI) is examined in thisstudy. Furthermore, this integra-tion scheme has another interesting property for metal forming. The phenomena of volumetriclocking is absent.The objective of this research is to examine the behavior of this nodal integration scheme.Therefore it is compared to a standard Gauss type of integration technique as used in finiteelements. Secondly the scheme is tested in incompressibility by simulating a footing problem.1W. Quak, A.H. van den Boogaard2 NUMERICAL SCHEMEThe SCNI method was introduced by Chenet al. [1]. The essence of the SCNI method isthat the strain at a node is determined by averaging the strain over a domain accompanying thatparticular node. This domain is usually a Voronoi cell. So informula form, the strain for smalldeformations is calculated as:ε̃ij(xk) =1Ωk∫Ωk12(∂ui∂xj+∂uj∂xi)dΩ (1)wherexk is the location of a node andΩk the accompanying area or volume for 2d or 3d re-spectively. This modified strain definition is used in a Galerkin weak form. For the followingtests, the displacement field is parameterized by local maxium-entropy shape functions asintroduced by Arroyo and Ortiz [2]. The shape functions are calculated by means of a Newton-Rhapson procedure. The tolerance to stop the iterations is the machine accuracy. The shapesposses first order reproducibility. The shape functions aredefined independently of the tessel-lation, which is only used for averaging the strain. Finally, the boundary conditions are appliedby using the method of Lagrangian multipliers. For now, the geometry is assumed linear.3 RESULTS3.1 a comparison of two integration schemesIn this test, the performance of the SCNI scheme is compared to a Gauss type integrationscheme. This integration scheme is constructed on the Delaunay triangulation of a cloud ofnodes. In each Delaunay triangle, a three point integrationrule is used. The SCNI integrationemploys strain averaging cells based directly on the Delaunay triangles. Instead of calculatingthe Voronoi diagram of the Delaunay triangulation, a triangle is divided in three parts and eachof the parts is used for the averaging procedure of the corresponding node. The construction ofVoronoi cells on the boundary is avoided this way. Constructing Voronoi cells at the boundaryof a domain is not straightforward. By applying the divergenc theorem on Equation (1) aboundary integral is obtained which is approximated with a tree point integration rule on eachof the cell sides.(a) beam model (b) triangulation and tesselationFigure 1: problem and discretizationFigure 1(a) shows the geometry of the problem. A cloud of nodes in the shape of the beamis generated. The nodal locations are randomly perturbed tomake an irregular grid. In Figure2W. Quak, A.H. van den Boogaard1(b) the computational geometrical discretization can be seen. At the left-hand side a part ofthe triangulation is displayed. The right-hand side shows the tesselation partly. Two load caseswere analyzed. Firstly the stretching of a plane strain beamis analyzed. Secondly the bendingof the beam is treated. A compressible linear elastic material model was used. Three runs wereperformed to examine the effect of the random perturbation on the accuracy. Figure 2 shows theresults. STD and SCNI are the abbreviations of the standard integration scheme and the nodalintegration scheme respectively. On the vertical axis a discrete error norm is plotted which isdefined as follows:eu =1N√√√√N∑i=1‖uh (xi) − uexact(xi) ‖2 (2)The nodal locations are given byxi, the approximated solution isuh, andN is the number ofnodes.10010110210-1010-810-610-410-2100SCNISTD√Neu(a) stretching10010110210-1010-810-610-410-2100SCNISTD√Neu(b) bendingFigure 2: erroreu for the two problemsIt can be seen that the behavior of the two schemes is different. SCNI is more accurate forthe first problem where the opposite is the case for the secondproblem. On the contrary, thestandard integration scheme gives similar performance forthe two problems. Secondly, it canbe seen that for the nodal integration, the error is not affected by the perturbation of the nodes.The error for the three simulations based on differently perturbed node grids is nearly equal.For integrating non-polynomial shapes the SCNI method seemsuitable, though a drawback isthe less accurate behavior in pure bending.4 footing of incompressible materialIn this test the performance of the method in incompressibility is examined. For this purposea square block of nearly incompressible material is supported on three sides and indented at the3W. Quak, A.H. van den Boogaardfree side. Figure 3(a) shows the geometry. The displacements of the left, bottom and right sideof the block are completely suppressed (stick). At the top side a pressure is prescribed to indentthe material. The test is performed with an elastic plane strain material model. Two cases areanalyzed, once with a Poisson’s ratioν f 0.499 and once withν = 0.49999. If there is locking,the results between the two cases would differ. The displacements withν = 0.49999 would bea lot smaller than the solution obtained withν = 0.499.(a) footing model (b) undeformed (c) ν = 0.499 (d) ν = 0.49999Figure 3: undeformed and deformed shapes of the footing problemFigure 3(b) shows the nodes of the undeformed geometry. Figures 3(c) and 3(d) show thedeformed configuration for a Poisson’s ratio ofν = 0.499 andν = 0.49999 respectively. It canbe seen that there is no spurious stiffening of the results. To check the absence of volumetriclocking for all circumstances a more rigorous test is required, though for now locking has notbeen found.5 CONCLUSIONThe comparison of the SCNI integration scheme with a standard integration scheme showeddifferences between the two. For the stretching of the beam,the SCNI scheme is more accuratethan the standard integration scheme whereas the pure bending of the beam showed the opposite.A footing test showed the absence of volumetric locking for the SCNI method. No spuriousstiffening was seen.Currently, the scheme is being implemented geometrically non-linear by means of an implicitupdated-Lagrangian formulation.REFERENCES[1] J.S. Chen, C.T. Wux, S. Yoon and Y. You. A stabilized conforming nodal integration forGalerkin mesh-free methods.Int. J. Num. Meth. Engng., 50, 435–466, (2001).[2] M. Arroyo and M. Ortiz. Local maximum-entropy approximation schemes: a seamlessbridge between finite elements and meshfree methods.Int. J. Num. Meth. Engng., 65,2167–2202, (2006).4

Nodal integration of meshless methods

A stabilized conforming nodal integration for Galerkin mesh-free methods.

Local maximum-entropy approximation schemes: a seamless bridge between ﬁnite elements and meshfree methods.

http://purl.utwente.nl/publications/69122

Nodal integration of meshless methods

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