Direct Reanalysis For Topological Modifications And Its Application

Efficient solution of finite element equations is a key technology as well as a bottleneck of CAD/CAE softwares.Reanalysis is a group of methods to efficiently solve structural responses with different kinds of modifications by using the solution information of the initial structure such as the factorization or result,which plays an important role in engineering problems such as design refinement,optimization and crack propagation etc.Among various problems,topological modifications which refer to structural layout change or remesh result in tremendous demand on topological reanalysis.In recent years,reanalysis has made progress in both approximate and direct methods.However,topological modification which involve addition/deletion of nodes and elements is still one challenge in sense of reanalysis.In literatures,approximate methods can quickly obtain an approximate response,but are usually limited to small-to-medium modifications.In contrast,direct methods deliver exact solution,but earlier implicit methods such as SMW show a poor efficiency.The newly developed block-based method achieves an improvement in efficiency while needs to preset the location of modifications.Moreover,most direct methods did not follow the state-of-art solution technologies and not applicable in practical softwares.Recently a method named UMTF(updating modified triangular factorization)based on the popular sparse solver was proposed and performed efficiently for local high-rank non-topological modifications.However,the indispensable feature of the sparse solver,i.e.the fill-in reduction state of matrices gets invalid when topological modifications are involved.Therefore,aiming at general topological modification,this paper develops a topological direct reanalysis by extending UMTF.The topological UMTF is exact,efficient and portable to current CAE softwares.The main work of this paper is as follows:1)Topological direct reanalysis algorithm with fixed background gridTo maintain the validity of the fill-in reduction state,a concept of integral structure which is composed of all possible nodes and elements is introduced to maintain an approximated fillin reduction state.Meanwhile,for matrix symbol changes,a memory-economic matrix shrinkage and extension in sense of the row sparse storage scheme are developed with negligible extra memory consumption.The algorithm is applicable to the construction sequence analysis and topology optimization with high efficiency.2)Topological direct reanalysis algorithm with adaptive background gridThrough analysis of the adverse effect of the topology modification on the fill-in reduction state,an update method of graph partition is proposed so that the fill-in reduction pivoting sequence is obtained along with the topological change.In addition,the matrix permutation scheme required by the graph partition update is supplemented.The algorithm is applicable for general topology modifications,such as crack propagation and local mesh refinement.3)Dynamic eigenvalue reanalysis algorithm of topological modificationsFinally,based on the subspace iteration method,a dynamic eigenvalue topological reanalysis algorithm is developed.The algorithm employs the topological UMTF to quickly obtain the factor matrix of the modified structure.Then,an initial iteration space is built on the basis of the original mode to reduce the iteration time,where the addition and deletion of DOF are taken into account.Combining the work in two phase,the eigenvalue topological reanalysis is developed.The topological UMTF algorithms are highly efficient for local high-rank modifications.The factor matrix is explicitly updated step-by-step.The algorithm can handle general topological modification,in which algorithms for problems with fixed background grid and adaptive-grid are respectively developed.The algorithms have little restrictions on modification,such as no preset of location is needed,only locality is required in terms of efficiency.Numerical examples show the algorithms can be used to million-scale problems with high-rank modifications.The algorithms can be used to practical engineering problems and have great value in applications.