Abaqus User Element implementation of NURBS based Isogeometric Analysis

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1 1 / 22 Abaqus User Element implementation of NURBS based Isogeometric Analysis 1, A. Duval 1, F. Maurin 1, H. Al Akhras 1 1 Université de Lyon, CNRS INSA-Lyon, Laboratoire de Mécanique des Contacts et des Structures, France 6 th European Congress on Computational Methods in Applied Science and Engineering Vienna, Austria Sep. 10 th - Sep. 14 th 2012

2 2 / 22 Outline and future work

3 3 / 22 Outline and future work

4 4 / 22 Isogeometric Analysis is based on the geometric primitives of CAD: NURBS and T-splines. It includes standard FEA as a special case, but offers other possibilities: Precise and efficient geometric modeling Simplified mesh refinement Smooth basis functions with compact support Superior approximation properties Accurate derivatives and stresses Integration of design and analysis Application in industry requires its implementation in commercial software

5 5 / 22 Several commercial FE packages provide user element capabilities. Abaqus User ELement and User ELement with abaqus MATerial subroutine offers the possibility to introduce arbitrary elements into abaqus using user defined material (UEL) or most of the material available in the software (UELMAT). UEL can be defined with arbitrary shape functions, node numbers, polynomial order and integration points. All the standard finite element data can be passed in the.inp input file. Additional data can be passed as common variables using User EXTERNAL DataBase (UEXTERNALDB) routines.

6 6 / 22 Outline and future work

7 Single patch implementation NURBS as a Basis for Analysis 95 Start Read input data Build connectivities and allocate global arrays K=0 and F=0 Loop through elements K e =0 and F e =0 Loop through quadrature points Evaluate basis functions and derivatives Add contributions to K e and F e Solve Kd=F Write output data Assemble K e K and F e F Stop 7 / 22 Figure 3.15 Flowchart of a classical finite element code. Such a code can be converted to a single-patch isogeometric analysis code by replacing the routines shown in green. J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs Isogeometric Analysis, 2009.

8 8 / 22 data Finite Element Analysis node coordinates element type element connectivity (ID, IEN and LM) integration rule NURBS based IGA control points coordinates polynomial degree element connectivity (ID, IEN, LM, INC) integration rule control points weights knot vectors Common data can be passed in the.inp abaqus input file. We only consider same polynomial order in all parametric directions, therefore element type defines the degree. Gauss quadrature is defined a priori based on element type.

9 9 / 22 Connectivity Given a basis function (i.e. control point) number and a parametric direction, the INC array returns the NURBS coordinate. Given a local basis function number and element number, the IEN array returns the global basis function number. Given a global basis function number and degree of freedom number, the ID array returns the equation number. The IEN array is directly passed in the.inp file with the standard element connectivity. The ID array is generated by abaqus based on the boundary conditions and number of DOF per CP.

10 9 / 22 Connectivity Given a basis function (i.e. control point) number and a parametric direction, the INC array returns the NURBS coordinate. Given a local basis function number and element number, the IEN array returns the global basis function number. Given a global basis function number and degree of freedom number, the ID array returns the equation number. The INC array is used in the pre-processing step to construct IEN but only a part of it is necessary in the shape function routine. For each element and parametric direction we only need the NURBS coordinate of the first local basis function. We replace INC by a new array called NIJK. Given an element number and a parametric direction, the NIJK array returns the NURBS coordinate of the first local basis function.

11 10 / Isogeometric Analysis: Toward Integration of CAD and FEA A multiple patch code A multi-patch isogeometricanalysiscodecanbemadetoconformwiththeflowchartin Figure In practice, however, it makes more sense to consider the slight modification shown in Figure In this case, we begin by inputting enough global information to build the global connectivities, as before. This information includes the polynomial orders and the knot vectors for each of the patches, but it does not require the control points. We can save time and memory by not reading the control points until they are needed. If local refinement Multi-patch implementation Start Read global input data Build connectivities and allocate global arrays K=0 and F=0 Loop through patches Read patch input data Loop through elements on the current patch K e =0 and F e =0 Loop through quadrature points Evaluate basis functions and derivatives Add contributions to K e and F e Solve Kd=F Write output data Assemble K e K and F e F Stop

12 which we insert into (3.58) along with = 0toarriveat w ƔD w ud + w fd + wudɣ = 0. (3.61) Though we have replaced one expression involving the unknown u for another, we have enforced the relationship between them weakly. 3.5 revisited Multi-patch implementation Local refinement ƔR wr dɣ β In Section 3 of Chapter 2 we discussed the need for modeling domains using multiple patches. We always assume compatible discretizations for the geometry, meaning that on the coarsest The outer mesh, mappings loop and onparameterizations patches cannot on the adjoining bepatch used faces are within identical. abaqus. Each control point on a face is in one-to-one correspondence For simplicity we only support C 0 with a control point from the adjoining face, likewise for the control variables of the solution. Incoupling many instances, with this relationship conforming will be meshes. preserved as we refine. To make the assembly of the stiffness matrices and force vectors as The connectivity simple as possible, the arrays connectivity forarray allwill patches identify the equivalent are generated local control variables in the on each face with a single control variable in the global array. By identifying them as a single pre-processing entity for analysis step. purposes, we simplify the logic and decrease the total amount of work needed. The result is that the two patches are joined as though they were one. Then a Identifying global two reordering control points atof the same thelocation elements in physical space andascontrol being a singlepoints entity is done is fairly trivial. Slightly subtler is what this implies for the basis functions. The situation is with a shown merging Figure 3.7. ofwhen CPtwo onbases patch generatedinterfaces. using open knot vectors are brought together, the effect is indistinguishable from the case of one knot vector with a knot repeated p times, The.inp as long file as the iscoefficients generated of the twofor joining the functions whole are thedomain same. This is, of ascourse, forexactly a single patch. what we have assured by identifying the two control variables as one. Patch data is passed in the.nb file. ƔR 10 / 22 Figure 3.7 If two knot vectors formed from open knot vectors are brought together, they can be made

13 11 / 22.inp file *HEADING **Job name:test.txt ** - Laboratoire de Mecanique des Contacts et des Structures - INSA-Lyon *Part, name=piece *USER ELEMENT, NODES=ncpelt, TYPE=Up, COORDINATES=d, variables=size of SVARS, INTEGRATION=ngelt, TENSOR=PSTRAIN or THREED 1,2 for 2D, 1,2,3 for 3D *Node,nset=AllNode ->list of control points cpnb, X-coordinate, Y-coordinate, Z-coordinate *Element, type=up, elset=allels -> list of elements eltnb, CP1, CP2, CP3,... *ELSET,ELSET=EltPAtch1 -> set of all the elements list of elements nb *NSET,NSET=SetCPFacei -> set of control points used for Dirichlet BC list of control points lying on the i-th face *ELSET, ELSET=SetEltFacei -> set of elements used for Neumann BC list of elements belonging to the i-th face *UEL PROPERTY, ELSET=EltPatch1, MATERIAL=MAT ->assign the material to the set of all elements *End Part ** ASSEMBLY *Assembly, name=assembly *Instance, name=i1, part=piece *End Instance *End Assembly

14 11 / 22.inp file **MATERIAL *MATERIAL,NAME=MAT *Elastic Young s modulus value, Poisson s ratio value *STEP,extrapolation=NO,NLGEOM=YES or NO *Static ** BOUNDARY CONDITIONS ->list of Dirichlet boundary conditions *Boundary I1.SetCPFacei, direction of the imposed BC, direction of the imposed BC, value imposed. ** LOADS ->list of Neumann boundary conditions *Cload I1.SetFacei, key for selecting element face, value imposed. ** OUTPUT REQUESTS *node file, frequency=1 U,RF,CF *el file, frequency=1 SDV *End Step

15 12 / 22.NB file *DIMENSION 2 or 3 * total nb of elements nelt *nb of patches nbpatch *nb of elements per patch neltp1, neltp2,... *data for patch i number of knots for parametric direction j knot vector for parametric direction j *NIJK array based on global domain element numbering eltnb, nijk1, nijk2, nijk3 *list of CP weights based on global domain CP numbering CPnb, weight

16 13 / 22 Procedure 1 Pre-processing 2 Analysis Matlab pre-processor based on Geopdes. Handles 2D, 3D single and multi-patch cases. Rhino plugin for multi-patch 2D cases (imposition of BC, material definition). 3 Post-processing

17 13 / 22 Procedure 1 Pre-processing 2 Analysis GUI for selecting IGA input files. Interactive job execution using our IGA user element. 3 Post-processing

18 13 / 22 Procedure 1 Pre-processing 2 Analysis 3 Post-processing Request output of the results in a raw ASCII file.fil (CP data, internal variables). GUI user interface for selecting input and output files and post-processing parameters. Abaqus python and C++ scripting interface to perform projection on a finer FE mesh. (Internal variables are first projected onto CP using global least-square if needed). Generate a projected output database binary file.odb readable in abaqus CAE.

19 Graphical User Interface movie 14 / 22

20 15 / 22 Outline and future work

: single patch horseshoe E= 100 000MPa, ν = 0.

21 : single patch horseshoe E= MPa, ν = 0.3, imposed displacement of ud = +/ 1ex. 16 / 22 T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, vol. 194, p , 2005.

22 17 / 22 : single patch horseshoe (cont d) σ zz on initial configuration σ zz on deformed configuration

23 18 / 22 : multipatch cylinder Cylinder submitted to internal pressure, geometry defined with 4 patches. R = 2m, r = 1m, E = MPa, ν = 0.3, p=10mpa. Von Mises stress on deformed configuration undeformed geometry and mesh.

Small strain plasticity: plane strain cylinder Cylinder submitted to internal pressure, elastic vs. perfectly plastic case. R = 2m, r = 1m, E=200000MPa, ν = 0.

24 Small strain plasticity: plane strain cylinder Cylinder submitted to internal pressure, elastic vs. perfectly plastic case. R = 2m, r = 1m, E=200000MPa, ν = 0.3, σ y=200mpa, p=145mpa. 19 / 22 Von Mises stress pure elastic case vm = 85MPa ; σvm MAX =335MPa) (σ min Von Mises stress perfectly plastic case (σvm min = 116MPa ; =200MPa) σ MAX vm

25 20 / 22 Outline and future work

26 21 / 22 We have a proposed an implementation of NURBS based IGA in the abaqus FEA commercial package. The implementation is able to handle single and multiple patches (C 0 continuity across compatible patch boundaries). Thank to UELMAT we can directly use most material available in abaqus for small and large deformation (NLGEOM). Post processing capabilities are available directly into abaqus CAE using Python/C++ scripting interface. Preliminary work on pre-processing allow us to provide a GUI for 2D cases and matlab for 3D.

27 22 / 22 Future work Improve the way necessary extra data can be passed on (i.e. outside the.inp file). Introduce Bezier extraction as an alternative. Include temperature for thermal or thermo-mechanical analysis, further test NLGEOM and material as well as dynamics. Released as open source under the CeCILL license in a few days.

Recent Developments in Isogeometric Analysis with Solid Elements in LS-DYNA

Recent Developments in Isogeometric Analysis with Solid Elements in LS-DYNA Liping Li David Benson Attila Nagy Livermore Software Technology Corporation, Livermore, CA, USA Mattia Montanari Nik Petrinic