Substructuring techniques—status and projections

Status and recent developments of substructuring techniques and their application to structural analysis and design are summarized. Discussion focuses on a number of aspects including: multilevel substructuring algorithms; use of hypermatrix and other sparse matrix schemes, use of substructuring in automated design systems and application of substructuring in elasto-plastic problems. Numerical examples are presented to demoostrate the reduction in the number of arithmetic operations and disk storage requirements obtained by using multilevel substructuring techniques. Also discussed are the potential benefits of using such techniques with new computing hardware such as CDC STAR-100 and minicomputer systems.     

Buckling analysis with the aska program system

The theoretical and practical considerations involved in extending the very large, general purpose finite element program system ASKA to include linear buckling mode analysis are presented. The main analysis features and their software implementation are discussed in the context of modular processors and data operands. A broader interpretation of buckling load factors and modes in relation to mode superposition is given, and typical applications are presented to illustrate both classical bifurcation buckling analyses and buckling mode superposition response.     

On nonlinear finite element analysis in single-, multi- and parallel-processors

Numerical solution of nonlinear equilibrium problems of structures by means of Newton-Raphson type iterations is reviewed. Each step of the iteration is shown to correspond to the solution of a linear problem, therefore the feasibility of the finite element method for nonlinear analysis is established. Organization and flow of data for various types of digital computers, such as single-processor/single-level memory, single-processor/two-level-memory, vector-processor/two-level-memory, and parallel-processors, with and without sub-structuring (i.e. partitioning) are given. The effect of the relative costs of computation, memory and data transfer on substructuring is shown. The idea of assigning comparable size substructures to parallel processors is exploited. Under Cholesky type factorization schemes, the efficiency of parallel processing is shown to decrease due to the occasional shared data, just as that due to the shared facilities.     

Nonlinear dynamics by mode superposition

A mode superposition technique for approximately solving nonlinear initial-boundary-value problems of structural dynamics is discussed, and results for examples involving large deformation are compared to those obtained with implicit direct integration methods such as the Newmark generalized acceleration and Houbolt backward-difference operators. The initial natural frequencies and mode shapes are found by inverse power iteration with the trial vectors for successively higher modes being swept by Gram—Schmidt orthonormalization at each iteration. The subsequent modal spectrum for nonlinear states is based upon the tangent stiffness of the structure and is calculated by a subspace iteration procedure that involves matrix multiplication only, using the most recently computed spectrum as an initial estimate. Then, a precise time integration algorithm that has no artificial damping or phase velocity error for linear problems is applied to the uncoupled modal equations of motion. Squared-frequency extrapolation is examined for nonlinear problems as a means by which these qualities of accuracy and precision can be maintained when the state of the system (and, thus, the modal spectrum) is changing rapidly.The results indicate that a number of important advantages accrue to nonlinear mode superposition: (a) there is no significant difference in total solution time between mode superposition and implicit direct integration analyses for problems having narrow matrix half-bandwidth (in fact, as bandwidth increases, mode superposition becomes more economical), (b) solution accuracy is under better control since the analyst has ready access to modal participation factors and the ratios of time step size to modal period, and (c) physical understanding of nonlinear dynamic response is improved since the analyst is able to observe the changes in the modal spectrum as deformation proceeds.     

On correction equations and domain decomposition for computing invariant subspaces

By considering the eigenvalue problem as a system of nonlinear equations, it is possible to develop a number of solution schemes which are related to the Newton iteration. For example, to compute eigenvalues and eigenvectors of an n × n matrix A, the Davidson and the Jacobi-Davidson techniques, construct ‘good’ basis vectors by approximately solving a “correction equation” which provides a correction to be added to the current approximation of the sought eigenvector. That equation is a linear system with the residual r of the approximated eigenvector as right-hand side.One of the goals of this paper is to extend this general technique to the “block” situation, i.e., the case where a set of p approximate eigenpairs is available, in which case the residual r becomes an n × p matrix. As will be seen, solving the correction equation in block form requires solving a Sylvester system of equations. The paper will define two algorithms based on this approach. For symmetric real matrices, the first algorithm converges quadratically and the second cubically. A second goal of the paper is to consider the class of substructuring methods such as the component mode synthesis (CMS) and the automatic multi-level substructuring (AMLS) methods, and to view them from the angle of the block correction equation. In particular this viewpoint allows us to define an iterative version of well-known one-level substructuring algorithms (CMS or one-level AMLS). Experiments are reported to illustrate the convergence behavior of these methods.     

Load dependent subspace reduction methods for structural dynamic computations

The evaluation of the dynamic response analysis of large structures by vector superposition requires in its traditional formulation the solution of a large and expensive eigenvalue problem. A new method of dynamic analysis using load dependent transformation vectors for systems subjected to fixed spatial distribution of dynamic loads was recently introduced by Wilson et al. as an economic alternative to the usual mode superposition method. New computational variants to generate a load dependent transformation basis for arbitrary transient loadings which are a function of space and time are presented. Numerical applications on a simple structural system are used to show the relative efficiency of the proposed solution procedure over classical solution methods using mode-displacement, mode-acceleration or the original (fixed) load dependent reduction method.     

Reduction techniques in dynamic substructures for large problems

Dynamic substructuring concepts find increasingly many applications in the solution of large dynamic problems. Substructure synthesis using fixed interface modes is found to be attractive from many angles, but has the major disadvantage in that a large number of interface or boundary degrees of freedom are carried over to system level thus increasing the cost of eigenvalue solution. Reduction of the number of these interface degrees of freedom is very essential for any cost effective solution, and some methods are suggested for this. With free vibration of the Indian Remote Sensing Satellite Structure as the base, reduction of interface degrees of freedom is first attempted by static condensation. It is found to give erroneous results even for frequencies, unlike its use in double Guyan's reduction method, where the results are generally satisfactory except for isolated modes. An alternate approach involving recursive or multilevel substructuring is suggested. The results obtained from the application of recursive substructuring clearly indicate that solution costs can be reduced without affecting the accuracy of results. Moreover, reduction of interface degrees of freedom by recursive substructuring makes it economical to use junction modes. This combination could be very effective in handling large problems.     

An efficient method for optimal structural design by substructuring

An optimal structural design technique incorporating the concept of substructuring in its formulation is presented. The method is developed using functional analysis techniques and the state space formulation of the optimal design problem. Design sensitivity analysis for the problem is discussed and an integrated computational algorithm for optimal design is presented. As an application of the method, optimal design of general trusses is presented. Numerical results for two standard truss structures of 25 and 200 members are obtained, using substructuring, and are compared with results obtained without substructuring. It is shown that the algorithm with substructuring is up to 66% more efficient.     

Structural topology optimization for frequency response problem using model reduction schemes

This study uses model reduction (MR) schemes such as the mode superposition (MS), Ritz vector (RV), and quasi-static Ritz vector (QSRV) methods, which reduce the size of the dynamic stiffness matrix of dynamic structures, to calculate dynamic responses and sensitivity values with adequate efficiency and accuracy for topology optimization in the frequency domain. The calculation of structural responses to dynamic excitation using the framework of the finite element (FE) procedure usually requires a significant amount of computation time; that is mainly attributable to repeated inversions of dynamic stiffness matrices depending on time or frequency intervals, which hastens the dissemination of the MR schemes in the analysis. However, using well-established MR schemes in topology optimization has not been prevalent. Therefore, this study conducted a comprehensive investigation to highlight the drawbacks and advantages of these MR schemes for topology optimization. In the results, the MS method, which generates reduction bases by considering some of the lowest eigenmodes, can lose the accuracy in both approximated structural responses and sensitivity values due to locally vibrating eigenmodes and higher mode truncation in the solid isotropic material with penalization (SIMP) approach. In addition, the RV and QSRV methods, which generate reduction bases by considering the external force, mass, and stiffness matrices of a structure, can be used as alterative model reduction schemes for stable optimization. Through several analysis and design examples, the efficiency and reliability of the model reduction schemes for topology optimization are compared and validated.     

A crankshaft system model for structural dynamic analysis of internal combustion engines

A system model for analyzing the dynamic behavior of an internal combustion engine crankshaft is described. The model couples the crankshaft structural dynamics, the main bearing hydrodynamic lubrication and the engine block stiffness using a system approach. A two-level dynamic substructuring technique is used to predict the crankshaft dynamic response based on the finite-element method. The dynamic substructuring uses a set of load-dependent Ritz vectors. The main bearing lubrication analysis is based on the solution of the Reynold's equation. Comparison with experimental results demonstrates the accuracy of the model. Numerical results also show the capabilities and significance of the model in engine crankshaft design.     

The cholesky method in substructuring with an application to fracture mechanics

The substructuring procedure proposed involves the modification of the Cholesky solution scheme. The required boundary stiffness matrix and equivalent load vector are obtained during the factorisation and forward substitution stage, respectively. The required alterations to the original algorithms are clearly shown, illustrating the minimal programming changes required. The implementation and ease of use is demonstrated in the finite element package, FINEL. An application to fracture mechanics is also provided, with the Virtual Crack Extension Method adapted to the Cholesky scheme and, in particular, the use in a substructuring environment. An example is presented demonstrating the use of the techniques proposed.     

Advances beyond transfer system dynamics compensation in real-time dynamic substructuring tests

Real-time dynamic substructuring tests, also known as hybrid tests, are critically affected by the dynamic nature of the actuators used to substitute the numerical substructure. These actuators are widely considered as transfer systems, which transfer magnitudes calculated via simulation of the numerical substructure to the physical substructure. Control engineering efforts in this area have, with few exceptions, traditionally been focused on compensation of the transfer system dynamics, in order to obtain perfect transfer systems. This article shows that the role of control systems in the context of real-time substructuring tests is not limited to transfer system dynamics compensation. Similarly, the role of actuators need not be that of transfer systems. As a result, a novel range of methods opens for actuator and control system design, which presents significant advantages over transfer systems with their corresponding dynamics compensation schemes.     

Computerized multiple level substructuring analysis

A computerized multiple level substructuring analysis method was developed and is presented. An ‘MCODE’ technique is introduced which forms an array in order to keep track of the degrees of freedom of the various matrices and vectors at all levels of substructuring. Constraints are introduced by use of the MCODE pointers also. By investigating the MCODE, the program can then position the rows and columns of the various matrices. A partitioned blocked data set method is used in the Cholesky decomposition for both the forward and backward substitutions, to reduce the I/O and CPU times. The sparseness of the stiffness matrix is taken into account to reduce the computer time during the multiplication step. The substructure connectivity information at every substructure level is input as data in user notation to allow very complex substructure cuts. The program checks for correct node point connection in space prior to any assemblage. A restart option is included to allow parallel engineering efforts on checking out substructures. An example case of a simply supported rectangular plate subjected to uniform lateral pressure is presented. Three runs, namely, without substructuring, with single level substructuring, and with three levels of substructuring were made and compared.     

Parallel processing for transient nonlinear structural dynamics of three-dimensional framed structures using domain decomposition

A strategy is presented for the solution of the fully nonlinear transient structural dynamics problem in a coarse-grained parallel processing environment. Emphasis is placed on the analysis of three-dimensional framed structures subjected to arbitrary dynamic loading and, in particular, steel building frames subject to earthquake loading. Concerns include long-duration dynamic loading, geometric and material nonlinearity, and the wide distribution of vibrational frequencies found in frame models. The implicit domain decomposition method described employs substructuring techniques and then a preconditioned conjugate gradient algorithm for the iterative solution of the reduced set of unknowns along the substructure interfaces. Substructuring is shown to provide a natural preconditioner for effective parallel iterative solution.     

Substructuring technique in nonlinear analysis of brick masonry subjected to concentrated load

A multi-level substructuring technique and a mesh grading scheme are used in the nonlinear finite element analysis of brick masonry subjected to in-plane concentrated loads. Masonry structures are ideally suited for solution using these techniques since masonry consists of a regular assemblage of bricks and joints in a repetitive pattern. Large wall panels can therefore be modelled without the need for excessive computer storage requirements. It is shown that the dual application of these techniques is highly efficient and leads to significant savings in costs. The possibility of modelling the elastic region of brick masonry as an isotropic continuum using similar techniques is also considered.     

A correct derivation of acceleration parameter for hopscotch and checkerboard (P,Q)-cyclic relaxation schemes

The correct method for applying the von Neumann stability analysis to composite finite difference schemes for numerical solution of partial differential equations is investigated. Our results provide justification of the hopscotch method and give correction to earlier analyses [1,7]. The methods employed here to analyze checkerboard and hopscotch iterative processes are also applicable to the study of more general composite (P, Q)-cyclic finite difference schemes.     

A numerical solution of two-dimensional fox-rabies dynamics of advection type

This article analyses a mode of two-dimensional fox-rabies dynamics with advection terms. The fox population is divided into two species: susceptible S and infective I. Finite-difference techniques are employed to compute the numerical solutions of this model. Iterative schemes with second-order accuracy are derived and their convergence is investigated by using the maximum principle analysis. An illustrative numerical example with graphical reis given to highlight the efficiency of the derived numerical schemes.     

视频字符叠加器叠加芯片比较

字符叠加（0sD）是一种在视频电子信号中叠加字符信息,使得电视图像中叠加有字符或汉字图形的技术。目前这项技术的应用极其广泛,因此,针对这种需求,结合市场最新技术对实现视频字符叠加进行研究与设计。文章论述了视频字符叠加的原理,并描述了目前市场上流行的叠加芯片的技术指标,就不同的芯片的不同的方案提出分析论证：着重了解MB90092、UPD6453和MAX7456方案的特性与使用方法。     

Energy load superposition and spatial optimization in urban design: A case study

Building energy consumption accounts for a large portion of total energy-use in a city or a regional district. However, energy load spatial distribution has seldom been considered during urban design phase. And energy conservation and energy efficiency measures pay more attention to individual building than buildings in a district or regional space as a whole. If buildings with different functions are mixed together and share same energy system, the savings on system capacity and peak electricity load can be significant. In this paper, a load superposition concept is proposed. The term ‘superposition’ refers to overlapping of energy demand load curves from different buildings and so that the total peak is smaller than the sum of individual peaks. Three spatial optimization methods of demand side load management and three different schemes of energy systems are proposed in this paper. And economic analysis is recommended to evaluate the different energy systems. The applicability of different approaches and the significance of load superposition was analyzed and elaborated through a case study to offer planners a feasible way for evaluating the potential of load spatial optimization.