A fully implicit numerical method,based upon a combination of adaptively refined hierarchical meshes and geometric multigrid,is presented for the simulation of binary alloy solidification in three space dimensions.The...A fully implicit numerical method,based upon a combination of adaptively refined hierarchical meshes and geometric multigrid,is presented for the simulation of binary alloy solidification in three space dimensions.The computational techniques are presented for a particular mathematical model,based upon the phase-field approach,however their applicability is of greater generality than for the specific phase-field model used here.In particular,an implicit second order time discretization is combined with the use of second order spatial differences to yield a large nonlinear system of algebraic equations as each time step.It is demonstrated that these equations may be solved reliably and efficiently through the use of a nonlinear multigrid scheme for locally refined grids.In effect this paper presents an extension of earlier research in two space dimensions(J.Comput.Phys.,225(2007),pp.1271–1287)to fully three-dimensional problems.This extension is validated against earlier two-dimensional results and against some of the limited results available in three dimensions,obtained using an explicit scheme.The efficiency of the implicit approach and the multigrid solver are then demonstrated and some sample computational results for the simulation of the growth of dendrite structures are presented.展开更多
The goal of efficient and robust error control, through local mesh adaptationin the computational solution of partial differential equations, is predicated on theability to identify in an a posteriori way those locali...The goal of efficient and robust error control, through local mesh adaptationin the computational solution of partial differential equations, is predicated on theability to identify in an a posteriori way those localized regions whose refinement willlead to the most significant reductions in the error. The development of a posteriori errorestimation schemes and of a refinement infrastructure both facilitate this goal, howeverthey are incomplete in the sense that they do not provide an answer as to where themaximal impact of refinement may be gained or what type of refinement — elementalpartitioning (h-refinement) or polynomial enrichment (p-refinement) — will best leadto that gain. In essence, one also requires knowledge of the sensitivity of the error toboth the location and the type of refinement. In this communication we propose theuse of adjoint-based sensitivity analysis to discriminate both where and how to refine.We present both an adjoint-based and an algebraic perspective on defining and usingsensitivities, and then demonstrate through several one-dimensional model problemexperiments the feasibility and benefits of our approach.展开更多
基金The authors would like to thank Jan Rosam and James Green for their contributions to the development of the implicit software used in this work.The PARAMESH software used in this work was developed at the NASA Goddard Space Flight Center and Drexel University under NASA’s HPCC and ESTO/CT projects and under grant NNG04GP79G from the NASA/AISR project.In addition we wish to thank Jonathan Dantzig for providing the explicit software used for comparison.The parallel simulations have been run on HECToR,the UK’s National Supercomputing Service,for which we gratefully acknowledge the compute time provided to us.
文摘A fully implicit numerical method,based upon a combination of adaptively refined hierarchical meshes and geometric multigrid,is presented for the simulation of binary alloy solidification in three space dimensions.The computational techniques are presented for a particular mathematical model,based upon the phase-field approach,however their applicability is of greater generality than for the specific phase-field model used here.In particular,an implicit second order time discretization is combined with the use of second order spatial differences to yield a large nonlinear system of algebraic equations as each time step.It is demonstrated that these equations may be solved reliably and efficiently through the use of a nonlinear multigrid scheme for locally refined grids.In effect this paper presents an extension of earlier research in two space dimensions(J.Comput.Phys.,225(2007),pp.1271–1287)to fully three-dimensional problems.This extension is validated against earlier two-dimensional results and against some of the limited results available in three dimensions,obtained using an explicit scheme.The efficiency of the implicit approach and the multigrid solver are then demonstrated and some sample computational results for the simulation of the growth of dendrite structures are presented.
基金The work of the third author was supported in part by NSF Career Award CCF0347791.
文摘The goal of efficient and robust error control, through local mesh adaptationin the computational solution of partial differential equations, is predicated on theability to identify in an a posteriori way those localized regions whose refinement willlead to the most significant reductions in the error. The development of a posteriori errorestimation schemes and of a refinement infrastructure both facilitate this goal, howeverthey are incomplete in the sense that they do not provide an answer as to where themaximal impact of refinement may be gained or what type of refinement — elementalpartitioning (h-refinement) or polynomial enrichment (p-refinement) — will best leadto that gain. In essence, one also requires knowledge of the sensitivity of the error toboth the location and the type of refinement. In this communication we propose theuse of adjoint-based sensitivity analysis to discriminate both where and how to refine.We present both an adjoint-based and an algebraic perspective on defining and usingsensitivities, and then demonstrate through several one-dimensional model problemexperiments the feasibility and benefits of our approach.
