Combination Of Approximation Research Articles

Structural static reanalysis using flexibility disassembly perturbation and system reduction

Rapid static reanalysis technology is an important tool commonly used in structural optimization design and reliability analysis. The existing static reanalysis methods are not efficient enough in solving high-rank large modification problems. To overcome this drawback, this paper proposes a new method for structural static reanalysis based on flexibility disassembly perturbation and system reduction. Firstly, a generalized flexibility disassembly perturbation technique is developed to directly compute the flexibility matrix of the modified structure. Subsequently, a new system reduction method is proposed to quickly approximate the exact solution of the modified static displacement. Even for the global large modification problem, the proposed method only requires two basis vectors to obtain very accurate calculation results. Two numerical models of statically indeterminate structures are used to verify the effectiveness and progressiveness of the method. The results show that the proposed reanalysis method has higher computational accuracy than the existing combined approximation (CA) and preconditioned conjugate gradient (PCG) methods, especially for the high-rank large modification case. A steel reticulated shell structure is further used to illustrate the application of the proposed reanalysis method in structural reliability analysis. It can be found that the proposed method significantly saves the calculation time of structural reliability analysis.

A structural reanalysis assisted harmony search for the optimal design of structures

Structural optimization usually requires a large amount of time-consuming structural analyses to evaluate the feasibility of the designs, which obstructs its application in large-scale problems. This study proposed a structural reanalysis-assisted harmony search (RAHS) algorithm for structural optimization. The RAHS utilizes the combined approximation (CA) technique instead of direct finite element analysis to evaluate the performances of the modified structures generated in iterations. Different from existing approaches, the proposed RAHS will update the reanalysis design point set and select the best reanalysis design point based on the similarity of the design vectors to improve the accuracy of the CA in a large search space. The RAHS also develops a diversity-based error criterion to accommodate the accuracy requirements of CA in different search phases of the harmony search. Five benchmark examples are utilized to examine the performances of the proposed RAHS. The results demonstrate that the Cosine distance outperforms the other four design point selection approaches, and the diversity-based error criterion can significantly reduce the computational cost. Compared with the penalty-based method and the improved Deb rule, the number of FE-based structural analyses can be lessened by about 90% and 50%, respectively, while yielding similar optimal designs.

Combined approximation based numerical vibration correlation technique for axially loaded cylindrical shells

As a widely used non-destructive buckling experimental technique, the vibration correlation technique (VCT) is able to determine the buckling load without reaching the collapse point. Following the experimental procedure, the numerical implementation of VCT can be easily realized. That is to say, the numerical VCT (NVCT) can be used for predicting buckling load. In this paper, the derivation of NVCT formulas is firstly presented for cylindrical shells under axial compression load. Aiming at accelerating the repeated eigenvalue analysis in NVCT, the combined approximation (CA) method is combined with NVCT. Then, the computational procedure of CA-NVCT is provided. Aiming at the verification of the effectiveness of the CA-NVCT, three analysis examples are carried out including variable-stiffness composite cylindrical composite shell, isotropic cylindrical shell and composite cylindrical shell with various types of imperfections. Finally, an optimization design is carried out to improve the buckling load of a composite cylindrical shell against imperfections based on the CA-NVCT. In comparison to results obtained by buckling experiments and typical buckling numerical methods, the high prediction efficiency and accuracy of the CA-NVCT in buckling analysis and optimization are verified.

A CA-XFEM for mixed-mode variable-amplitude fatigue crack growth

In this research, the extended finite element method (XFEM) in conjunction with the combined approximation (CA) is utilized for estimating the fatigue life of two-dimensional isotropic bodies under variable-amplitude loading. In the proposed method, called CA-XFEM, in addition to the fact that no re-meshing process is required, the crack growth path is determined without the need to solve the whole system of equations which these features significantly reduce computational costs. The Willenborg model is employed for modeling the retardation effect due to the overloads in the load history. For validation of the method, the numerical results are compared with the existing experimental data for a compact tension specimen made of Al 7075-T6 and a compact tension shear specimen made of Al 5083-H111. It is observed that the developed CA-XFEM Matlab code has excellent capability in modeling variable-amplitude fatigue crack propagation considering the retardation effect. Also, the effect of ratio and sequence of overload and mixed-mode overloading on the fatigue crack growth are studied.

Structural Dynamic Modification Based on Combined Approximations Method

Structural dynamic modification plays an important role in structural dynamic design. This paper presents an algorithm for dynamic modification of the structure, which is based on the Combined Approximations (CA) approach, sensitivity analysis, Taylor series expansion and the least square method. The feasibility of the algorithm is verified by a numerical example and the results show that the algorithm is accurate enough and easy to be implemented.

A discontinuous isogeometric reanalysis method and its application in closed-loop optimization problems

In this study, a DisContinuous IsoGeometric Reanalysis (DCIGR) method is proposed to analyze and optimize the model of Non-Uniform Rational B-Splines (NURBS) patches with non-matching interfaces. In the DCIGR, the Nitsche method is adapted to couple NURBS patches with gaps and the smooth algorithm is developed to correct singular stresses resulting from gaps. However, the additional stiffness matrices (interfacial stiffness matrices) must be calculated when using the Nitsche method, which will inevitably increase the computational cost. Therefore, Combined Approximation (CA) is introduced to reduce the computational cost of the optimization. Theoretically, the larger and more complex a problem is, the more efficient the DCIGR will be. Moreover, since the modifications of the model can be accurately represented by the IGA, the proposed method can be easily integrated with the closed-loop optimization. Compared with existing methods for non-matching interfaces, the DCIGR fully takes the advantage of the IGA and achieves the fast optimization based on the CA. Numerical examples are given to verify the accuracy and efficiency of the proposed method.

Accelerated Koiter method for post-buckling analysis of thin-walled shells under axial compression

Due to the high specific strength and stiffness, thin-walled shells are widely used in aerospace engineering structures. However, along with the increase of the structure size or the structure hierarchy, the computational cost of the post-buckling analysis of thin-walled shells would increase sharply and then the imperfection sensitivity analysis would become time-consuming. In this study, an accelerated Koiter method is proposed to improve the efficiency of post-buckling analysis and imperfection sensitivity analysis of thin-walled shells under axial compression. In the framework of the accelerated Koiter method, a single mode Koiter method considering pre-buckling nonlinear behavior is used to reduce the computational time of repeated imperfection analysis, which can accurately predict the effect of the small amplitude imperfection. Furthermore, in order to obtain the expanded point of Koiter method, a Combined Approximation (CA)-based iterative eigenvalue algorithm is constructed to obtain the pre-buckling state in the neighborhood of the bifurcation point and reduce computational cost by the reasonable step length prediction. Finally, the efficiency and accuracy of the proposed method are demonstrated by means of four illustrative examples, including the composite cylindrical shell, the isotropic cylindrical shell, the isotropic conical shell and the hierarchical stiffened cylindrical shell under axial compression.

A fast reanalysis solver for 3D transient thermo-mechanical problems with temperature-dependent materials

A fast reanalysis solver using linear tetrahedral elements is developed for 3D transient thermo-mechanical problems with temperature-dependent materials. The core idea is to formulate the reanalysis framework of a stable node-based smoothed finite element reanalysis method. In the developed solver, the stable node-based smoothed finite element method (SNS-FEM) is used as the main solver due its accuracy and stability. The combined approximations (CA) method is the auxiliary tool for improving the efficiency of the SNS-FEM. In the developed reanalysis framework, binomial series expressions are used as high-quality basis vectors for the reduced basis method. The transient temperature and displacement fields can then be calculated without solving the complete set of nonlinear or linear system equations. The performance of the developed solver is evaluated through several numerical examples with different kinds of mixed boundary conditions. The results show that the reanalysis framework with two or three basis vectors is 20–50 times faster than the complete analysis and that the solver is much more efficient and stable than other solvers that use the traditional finite element method (FEM) or the smoothed finite element method (NS-FEM). Therefore, the performance of the developed solver is demonstrated.

Improved Woodbury approximation approach for inelasticity-separated solid model analysis

Solid elements are widely implemented in conventional finite element method (FEM), but they also consume more computing resources relative to the bar, beam and shell elements because of the high dimensional tangent stiffness matrix recalculation and factorization. In this work, an inelasticity-separated solid element model and an efficient numerical solution procedure are proposed for the material nonlinear analysis of three-dimensional (3D) entity structures. This solid element model was developed within the framework of the inelasticity-separated finite element method (IS-FEM) presented in prior studies [1], the computational efficiency of IS-FEM is better than that of conventional FEM for the local material nonlinearity problem when using the Woodbury formula. The tangent stiffness matrix of a structure is expressed by the sum of the invariant global stiffness and another changing Schur complement modification with a low-rank. However, a nonlinearity problem with solid elements may have more inelastic degrees of freedom (IDOFs), thus, the efficiency of IS-FEM is low due to the high-rank Schur complement modification. This study improved the solution procedure of the Woodbury approximation Method (WAM), which applies the combined approximation (CA) approach within the framework of the Woodbury formula. The new derived solution scheme only contains limited back-substitutions of the global initial stiffness matrix and product of the sparse matrix and vector during a certain iteration step, which completely avoids the Schur complement matrix factorization and the constant matrix precalculation and storage process. Time complexity analysis indicates that the efficiency of the improved methodology is greatly enhanced relative to both the conventional FEM and the IS-FEM with the exact Woodbury formula, and the ratio of the efficiency improvement increases with an increase in the degrees of freedom (DOFs). Finally, the numerical examples demonstrate the validity and efficiency of the presented solid element model and the improved solution procedure.

An exact and efficient X-FEM-based reanalysis algorithm for quasi-static crack propagation

• A specific and exact reanalysis algorithm is suggested to analysis crack propagation problem based on X-FEM. • Compared with the popular reanalysis algorithm, the displacement and stress can be obtained more accurately. • Local stiffness matrix updating strategy is suggested to achieve the modified stiffness matrix . • Cholesky factorization of stiffness matrix allows efficient updating. • Compared with classical X-FEM, the efficiency of suggested method is significantly improved. This study suggests a specific reanalysis algorithm termed decomposed updating reanalysis (DUR) for quasi-static linear crack propagation based on the extended finite element method (X-FEM). It is well known that the number of iterative steps is usually very large during X-FEM simulation procedures because a small crack increment is required to improve the accuracy of the simulation. However, according to the features of the X-FEM, the small crack increment only influences the nearby elements and only leads the local change of the stiffness matrix at each iterative step. Therefore, the DUR method is proposed to accelerate the X-FEM solving process by only calculating the changed part of the equilibrium equations. Moreover, the local updating strategy can efficiently update the modified stiffness matrix and the Cholesky factorization. Compared with other reanalysis algorithms, such as combined approximations (CA), the DUR method is more accurate. Numerical examples demonstrate that the DUR method improves the efficiency of the X-FEM significantly with a high accuracy.

Fast sensitivity reanalysis methods assisted by Independent Coefficients and Indirect Factorization Updating strategies

In this study, the novel sensitivity reanalysis methods are suggested to obtain the static displacement sensitivity to local modifications efficiently. Details of displacement sensitivity formulations are deduced from the displacement reanalysis equations of Independent Coefficients (IC) and Indirect Factorization Updating (IFU) strategies. Moreover, the performances of the proposed sensitivity reanalysis methods are investigated to compare with Combined Approximation (CA) methods. Different numerical examples and efficient comparison are performed to confirm the accuracy and efficiency of the proposed methods. As a result, the proposed sensitivity reanalysis methods are proved to be more efficient with large scale problems and the accuracy of modified Degrees of Freedoms (DOFs) can be guaranteed as well.

An accurate and efficient algorithm for the simulation of fatigue crack growth based on XFEM and combined approximations

This study aims to reduce the computational cost in the simulation of fatigue crack growth process. Extended finite element method and combined approximations (CA) are integrated to form an efficient algorithm for such analysis. In the CA approach, binomial series are used as high quality basis vectors for the reduced basis expression and then crack propagation path can be predicted without solving the complete set of system equations. The validity of presented algorithm is fully investigated through several numerical examples. From these results, it is shown that the presented algorithm is very accurate and can save huge amounts of computational effort.

Extension of reanalysis method, analysis of heat conduction by using CA and IC methods

Two fast heat transfer solvers by using Node-based Smooth FEM method (NS-FEM) and reanalysis methods are suggested. The reanalysis is an auxiliary solver which can predict the response of a given problems instead of full analysis. In this study, combined approximations (CA) and independent coefficients (IC) methods are investigated. The CA might be the most popular reanalysis method and has been widely applied to structural analysis. In the CA method, binomial series is used as high quality basis vectors for reduced basis expression. The IC method is suggested to reanalyze structures with local modifications which lead to a low-rank change in the stiffness matrix. Steady temperature fields can be calculated without solving a complete set of linear system equations by both of them. Compared with the CA method, the IC method requires only the initial solution as input and can determine the independent coefficients for each degree of freedoms (DOFs) influenced by structural modifications without matrix decomposition operation. In order to verify the performances of the CA method and the IC methods, several numerical examples are tested. The results demonstrate that the CA and the IC methods have high accuracy as well as efficiency when the modification is local.

A novel PCA-based microstructure descriptor for heterogeneous material design

A novel PCA (Principal Component Analysis)-based microstructure descriptor is proposed for heterogeneous material design given a database of material microstructures. The PCA-based descriptor captures the important geometric features of the database, and provides a compact representation of the design space. Based on the PCA-based descriptor, a polynomial expression is ultimately derived that links the microstructures in the design space with their associated physical performances. Compared with previous approaches using first-order or second-order descriptors, the novel approach integrates PCA and Combined Approximation (CA) in mechanics, provides more precise presentation of the microstructure and accelerates the optimization and reconstruction of the result microstructure. The performance of the proposed approach is also tested in various aspects for a linear elasticity analysis problem.

A Parallel Elastoplastic Reanalysis Based on GPU Platform

An efficient parallel elastoplastic reanalysis method is suggested. The main backbone of the suggested method is based on combined approximation (CA) reanalysis. GPU parallel computation is used to accelerate assembling the stiffness matrix. Assembling process is divided into the offline part for strain matrix and online part for element stiffness matrix, which makes the structure of the program more reasonable and efficient. Pseudo elastic analysis is introduced and extended to load increment method to make the CA method more feasible. The numerical examples show that the suggested method can improve the efficiency of elastoplastic analysis significantly and the accuracy of results can also be ensured.

Sensitivity reanalysis of vibration problem using combined approximations method

Sensitivity is indispensable to structural modification and optimization. This paper focuses on the analytical sensitivity reanalysis for vibration problem in the framework of combined approximations (CA) method. The sensitivity reanalysis formulations of eigenvalues and eigenvectors are derived from the vibration equation reduced by CA method, where the eigenvector sensitivity is solved by Nelson's method. Numerical examples demonstrate the accuracy and efficiency of the proposed reanalysis method. Especially, this method can greatly improve the efficiency of sensitivity analysis and can accelerate the gradient-based structural optimization constrained with frequencies and modal shapes.

Fast and efficient analysis of transient nonlinear heat conduction problems using combined approximations (CA) method

The combined approximations (CA) method is very efficient and can provide high quality results, when dealing with structural optimization problems. In this study, the CA method is utilized to reduce the computational effort in transient nonlinear heat transfer analysis. In the CA approach, binomial series are used as high quality basis vectors for reduced basis expression. Transient nonlinear temperature fields are then calculated without solving complete set of nonlinear system equations. Both two-dimensional and three-dimensional numerical problems are studied to verify the accuracy and efficiency of CA method, when solving transient nonlinear heat conduction problems.

A Kind of Optimization Method on Plate-Shell Structures with Stiffeners

An new optimization method which combines layout optimization of stiffeners with structural parameters optimization of structures is discussed. The first step is to optimize the layout of stiffenerser, then the structural parameters were optimized. In layout optimization of stiffeners, the strain energy density sensitivities of the elements were used to determine which elements of stiffeners should be deleted. In structural parameters optimization, the object function and the constraint functions were approximated by the second-order Taylor expansion. DFP (Davidon, Fletcher and Powell) method was presented to solve optimal problem. In order to reduce computational effort, the combined approximation (CA) method was used to reanalysis and update the displacements and stresses of the structure. The present method was applied to a bunker. The numerical results show that the optimization method was effective for optimization of plate-shell structure with stiffeners, and it could be to implement on a computer.

Shift-Relaxation Combined Approximations Method for Structural Vibration Reanalysis of Near Defective Systems

Structural reanalysis for defective systems and the near defective systems is discussed in this paper. Specially, by a relaxation factor embedded in the Combined Approximations (CA) approach and a frequency shift, the reanalysis problem of near defective systems with close eigenvalues can be transformed into one of defective systems with repeated ones, which is equal to the average of the close ones. Numerical examples show that the proposed method is effective and stable in reanalysis problem of near defective systems.

A Parallel Reanalysis Method Based on Approximate Inverse Matrix for Complex Engineering Problems

The combined approximations (CA) method is an effective reanalysis approach providing high quality results. The CA method is suitable for a wide range of structural optimization problems including linear reanalysis, nonlinear reanalysis and eigenvalue reanalysis. However, with increasing complexity and scale of engineering problems, the efficiency of the CA method might not be guaranteed. A major bottleneck of the CA is how to obtain reduced basis vectors efficiently. Therefore, a modified CA method, based on approximation of the inverse matrix, is suggested. Based on the symmetric successive over-relaxation (SSOR) and compressed sparse row (CSR), the efficiency of CA method is shown to be much improved and corresponding storage space markedly reduced. In order to further improve the efficiency, the suggested strategy is implemented on a graphic processing unit (GPU) platform. To verify the performance of the suggested method, several case studies are undertaken. Compared with the popular serial CA method, the results demonstrate that the suggested GPU-based CA method is an order of magnitude faster for the same level of accuracy.