We have modified a 3D hydrodynamics code so that it has the capability to impose periodic boundary conditions on the problem being considered. This capability allows it to treat only a basic symmetry unit of the problem when translational or rotational periodic symmetries are present. The code has been run successfully for two test problems involving rotational symmetries

Morgan, D L

Neely, J R

Vantine, H C

UNT Digital Library

September 18, 1998This is a preprint of a paper intended for publication in a journal or proceedings.  Sincechanges may be made before publication, this preprint is made available with theunderstanding that it will not be cited or reproduced without the permission of theauthor.LawrenceLivermoreNationalLaboratoryUCRL-JC-132010PREPRINTPeriodic Boundary Conditions in a 3D Hydro CodeThis paper was prepared for submittal to theModern Trends in Computational PhysicsDubna, RussiaJune 15-20, 1998D. L. Morgan, Jr., J. R. Neely, H. C. VantineDISCLAIMERThis document was prepared as an account of work sponsored by an agency ofthe United States Government.  Neither the United States Government nor theUniversity of California nor any of their employees, makes any warranty, expressor implied, or assumes any legal liability or responsibility for the accuracy,completeness, or usefulness of any information, apparatus, product, or processdisclosed, or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial product, process, or service by tradename, trademark, manufacturer, or otherwise, does not necessarily constitute orimply its endorsement, recommendation, or favoring by the United StatesGovernment or the University of California.  The views and opinions of authorsexpressed herein do not necessarily state or reflect those of the United StatesGovernment or the University of California, and shall not be used for advertisingor product endorsement purposes.Periodic Boundary Conditions in a 3D Hydro CodeDavid L. Morgan, Jr., J. Robert Neely, and Harry C. VantineLawrence Livermore National Laboratory,  Livermore, California  94551,  U.S.A.morgan6@llnl.govAbstractWe have modified a 3D hydrodynamics code so that it has the capability toimpose periodic boundary conditions on the problem being considered. Thiscapability allows it to treat only a basic symmetry unit of the problem when trans-lational or rotational periodic symmetries are present. The code has been runsuccessfully for two test problems involving rotational symmetries.IntroductionWhen symmetries are present in hydrodynamics problems, the computational effortrequired to solve them can be substantially reduced. While most hydrodynamics com-puter codes can deal with reflective symmetries, it is less common that they can handleperiodic symmetries through the use of periodic boundary conditions. Recently we haveadded this latter capability to a 3D, finite element, time-explicit hydrodynamics codecapable of running on parallel computers. At present the capability is only available forpure Lagrangian problems, but when fully implemented to run with advection present, itwill enable the code to solve efficiently problems such as jet-engine blade simulation,where it need include only a single blade in the calculation instead of all blades equallypositioned around an axis.Periodic boundary conditions arise when the symmetries involved in a problem aresuch that it can be fully described as consisting of two or more units which are identical inall of their features except for position and orientation. Under these circumstances, eachsmallest possible unit (basic symmetry unit) is a volume (in 3D) surrounded by a numberof surfaces, each of which is a member of a pair within which the two surfaces are identi-cal except for position and orientation. They are identical in shape as well as in the distri-butions upon them of intrinsic quantities, such as pressure, density, and velocity.Periodic boundary conditions consist of requiring that these identities are present in eachbasic unit. When the conditions are applied, it is then necessary to solve the problem foronly a single such unit.Periodic conditions can exist in conjunction with reflective conditions, in which casethe basic symmetry units for each may or may not be the same. If they are the same, theresults of applying either type of boundary condition will be the same, but applying reflec-tive conditions is simpler.ImplementationIn the code, all hydrodynamic quantities are numerically represented by, or arederived from, the positions of a set of points in three dimensional space, called nodes,that form the mesh for the numerical model of the problem. These quantities include thenode-centered quantities, position and velocity, as well as the element-centered quanti-ties such as energy, density, and stress, whose values ultimately depend on the positionsof the eight nodes surrounding the element. The motion of these nodes defines themotion of the material as time progresses.Periodic boundary conditions are implemented by first requiring the user to specifythe pairs of boundaries (surfaces) that are periodically symmetric. This is done by spec-ifying their nodesets. A nodeset is simply a subset of the nodes in the mesh and can bedefined either during mesh generation, which occurs before running the hydrodynamicscode, or by using input commands built into the code itself. The user must, in addition,supply a description of how to translate and/or rotate the first nodeset of each pair so thatit will overlay the second. It is required that after this translation and rotation, the nodesin each set match up one-to-one.The motion of each node is determined by a force calculation. During this calcula-tion, forces on the nodes on each boundary contribute to the forces on each of the corre-sponding nodes on the other boundary of each pair. If the boundaries are such that theyare rotated relative to each other, then the force vector must be rotated as well beforeadding in the contribution. This exchange of partial forces between boundaries ensuresthat the boundaries will maintain the same shape relative to each other, differing only bytheir rotational relationship in 3D space.In some problems, the boundaries in a symmetric pair intersect with each other. Inthe test problems described below, the periodic boundaries are 90 degrees to eachother, with a 1D line of nodes which falls into both nodesets. Force calculations on thesenodes are handled separately from the other non-intersecting nodes. It is required thatthe angle, theta, between the nodesets on either side of the intersection is such that itdivides evenly into 360 degrees, as is the case for rotational periodic symmetry. Thenthe force vector is rotated and added to the original N times, where N is 360/theta. Forexample, the force along two of the intersections in the problem described below is F +F_90 + F_180 + F_270, where F_r is the original force, F, rotated around r degrees. This,of course, results in a net force of zero perpendicular to the axis of rotational symmetry(such axes are invariant) and provides a check on the accuracy of the code.The implementation is fully parallelized, and the test problems described here wereall run on the ASCI Blue Pacific machine at Lawrence Livermore National Laboratoryusing 32 processors.TestingWe have tested the 3D periodic boundary condition capability by applying it to aproblem consisting of an infinite volume of high explosive detonated by an infinite, repeti-tive array of lighting points. The basic symmetry units here are initially cubes. This highenergy test case was chosen so that significant surface distortions would occur. Twolighting-point arrays were used, single layers of which are described in Figures 1 and 2.The array of cubes is shown in Fig. 3. The first lighting array is such that only periodicconditions are present. With the second, both periodic and reflective conditions arepresent, but some of the symmetry units are different.In the test with the first array, ALE3D demonstrated the required surface identities atall times. Those results are not shown here because they are embodied in the results ofthe test with the second array. In that test the effects of periodicity were precisely repro-duced (Figures 4 and 5), and even though only periodic boundary conditions wereemployed, the results show the reflective symmetries that exist in the problem as accu-rately as possible with a mesh which does not itself possess those symmetries (Figures5 and 6).ConclusionWe have successfully added a periodic-boundary-condition capability to our 3DLagrangian hydrodynamics code. Work is in progress to generalize this capability to aversion of the code that includes advection so that we can solve problems involving flowthrough the surfaces of the basic symmetry unit.AcknowledgmentThis work was performed in part by Lawrence Livermore National Laboratory underthe auspices of the US Department of Energy under contract W-7405-ENG-48.Keywords3D, hydrodynamics, boundary conditions, periodicFig. 1. Positions of lighting points (dots) in one particular x-y plane for the testsof the periodic-boundary-condition capability. Only the lighting points at thesolid dots were employed in the first test. Such an array has no reflective symme-tries. The large squares (two are highlighted on the left) are basic units of trans-lational periodic symmetry for this array. When rotational periodic symmetry isconsidered, the basic units are smaller ones, some of which are shown on theright. Their symmetries are shown in Fig. 2. In the second test, lighting points atthe positions of the open dots were included. This adds a number of obviousreflective symmetries while retaining all periodic ones. (Also added are moreperiodic symmetries. Of these, the rotational ones are around the corners, cen-ters, and edge centers of each small square of the array of lighting points. Thesewere not considered, but their effect can be seen in Figures 4., 5., and 6.)xyFig. 2. The basic rotational periodic symmetry units (dashed edges) in an x-yplane containing lighting points (dots) for either array described in Fig. 1. Therotations indicated will move the highlighted unit into coincidence with neighbor-ing units. A result of this is that all quantities on side A are identical to those onside D (after rotation of the vector quantities like velocity) at corresponding loca-tions.  The same is true for sides B and C.Fig. 3. Geometry for the test problems. The lighting points are shown as theywere for the first test.ab cdABCDa,c   180o rotational symmetryb,d 90o rotational symmetryD = A  rotated 90 o about dB = C  rotated 90 o about bxxxyLighting pointUnit CellInfinite Array(cube)High ExplosivexyzFig. 4. A grey-scale intensity plot of material speed (cm/ µs) in the second test ofthe code at a time when all of the high explosive has detonated with the exceptionof a small portion near the center of the figure. The view is of the two surfaces ofthe now distorted cube which correspond to lines B (left surface) and C (right sur-face) of Fig. 2. The periodic symmetry (invariance under rotation by 90 degreesabout the near, vertical edge) requires that the left and right portions of the viewbe mirror images of each other, except for small variations due to distortions fromflatness of the cube’s faces. This is seen to be the case. Note that each verticaledge of the cube is an axis of rotational symmetry; thus they are invariant, and itmay be seen that they have remained vertical and straight. Note also the reflec-tive symmetry (unimposed) about the mid-z plane which results from the fact thatthe lighting point patterns are identical and equally separated in the z direction.cycle 120Fig. 5. A grey-scale intensity plot of material speed (cm/ µs) at the same time as inFig. 4. The distorted cube has been rotated relative to the view in Fig. 4 so thatthe top may be seen.cycle 120Fig. 6. A grey-scale intensity plot of material speed (cm/ µs) at the same time as inFig. 4. The view is of the upper surface (maximum z coordinate) looking towardnegative z. The upper edge and right edge are the 90 degree rotations of oneanother as are the left and lower edges, thus demonstrating the imposed rota-tional periodicity. The pattern shows the unimposed (but present) reflective sym-metry about various lines with y-x slopes of -2 and 1/2 as expected from thedetonation pattern employed in the second test of the code.cycle 120

Periodic boundary conditions in a 3D hydro code

https://digital.library.unt.edu/ark:/67531/metadc688194/m2/1/high_res_d/3646.pdf

Periodic boundary conditions in a 3D hydro code

Abstract

Similar works

Full text

Available Versions

UNT Digital Library