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NASA NTRS1976

Free-interface methods of substructure coupling for dynamic analysis

Benfield, et al. (1972) showed that among fixed-interface, free-interface, and hybrid substructure coupling methods, the fixed-interface methods as the most accurate and the free-interface methods are the least accurate. In the present note, a substructure coupling method is proposed which employs free-interface substructure modes supplemented by 'reduced flexibility.' Substructure coupling based on the improved substructure model is discussed, and a numerical comparison with Hou's free-interface method is given. To simplify representation of the method proposed, the substructure equations are developed first for constrained substructures, and then the equations representing substructures with rigid-body modes are given. Finally, the equations for coupling of substructures are derived. Example calculations are included.

Craig, R. R., Jr.↗

NASA NTRS1971

Substructuring techniques

The substructure function generator program is discussed. Input to this program consists of a definition of a finite element model of a substructure, and specification of the type and number of displacement functions to be generated. Primary output is a substructure data file containing the substructure mass and stiffness matrices expressing kinetic and potential energies as quadratic forms in coefficients of the displacement functions, etc. The substructure synthesis program forms complete system mass, stiffness, and damping matrices, computes system modes and frequencies, and executes transient response calculations. Input to this program consists of the array of substructure data files generated by the function generator program for individual substructures, and data cards defining the position and interconnection of the substructures, damping data, forcing function details, and function control parameters.

Whetstone, W. D.↗

NASA NTRS1984

A substructure coupling procedure applicable to general linear time-invariant dynamic systems

A substructure synthesis procedure applicable to structural systems containing general nonconservative terms is presented. In their final form, the non-self-adjoint substructure equations of motion are cast in state vector form through the use of a variational principle. A reduced-order model for each substructure is implemented by representing the substructure as a combination of a small number of Ritz vectors. For the method presented, the substructure Ritz vectors are identified as a truncated set of substructure eigenmodes, which are typically complex, along with a set of generalized real attachment modes. The formation of the generalized attachment modes does not require any knowledge of the substructure flexible modes; hence, only the eigenmodes used explicitly as Ritz vectors need to be extracted from the substructure eigenproblem. An example problem is presented to illustrate the method.

Howsman, T. G.↗

NASA NTRS1984

A substructure coupling procedure applicable to general linear time-invariant dynamic systems

A substructure synthesis procedure applicable to structural systems containing general nonconservative terms is presented. In their final form, the nonself-adjoint substructure equations of motion are cast in state vector form through the use of a variational principle. A reduced-order mode for each substructure is implemented by representing the substructure as a combination of a small number of Ritz vectors. For the method presented, the substructure Ritz vectors are identified as a truncated set of substructure eigenmodes, which are typically complex, along with a set of generalized real attachment modes. The formation of the generalized attachment modes does not require any knowledge of the substructure flexible modes; hence, only the eigenmodes used explicitly as Ritz vectors need to be extracted from the substructure eigenproblem. An example problem is presented to illustrate the method.

Howsman, T. G.↗

NASA NTRS1973

NASTRAN multipartitioning and one-shot substructuring

For intermediate size problems where all the data is accessible, the present method of substructuring in three separate phases (for static analysis) is unneccessarily cumbersome. The versatility of NASTRAN's DMAP and internal logic lends itself to finding a practical alternative to these procedures whereby self-contained special-purpose ALTER packages can be written to be run in one pass. Two examples are presented here under the titles of multipartitioning and one-shot substructuring. The flow of multipartitioning resembles that of the present three-phase substructuring. The basic effect is to partition the structure into substructures and operate on each substructure separately. This can be used to reduce the bandwidth of a given problem as well as to store information which will allow a change to be made in one of the substructures in a later run. This latter procedure is carried out in a second program titled one-shot substructuring.

Levy, A.↗

NASA NTRS1995

Parallel Computational Environment for Substructure Optimization

Design optimization of large structural systems can be attempted through a substructure strategy when convergence difficulties are encountered. When this strategy is used, the large structure is divided into several smaller substructures and a subproblem is defined for each substructure. The solution of the large optimization problem can be obtained iteratively through repeated solutions of the modest subproblems. Substructure strategies, in sequential as well as in parallel computational modes on a Cray YMP multiprocessor computer, have been incorporated in the optimization test bed CometBoards. CometBoards is an acronym for Comparative Evaluation Test Bed of Optimization and Analysis Routines for Design of Structures. Three issues, intensive computation, convergence of the iterative process, and analytically superior optimum, were addressed in the implementation of substructure optimization into CometBoards. Coupling between subproblems as well as local and global constraint grouping are essential for convergence of the iterative process. The substructure strategy can produce an analytically superior optimum different from what can be obtained by regular optimization. For the problems solved, substructure optimization in a parallel computational mode made effective use of all assigned processors.

Gendy, Atef S.↗

NASA NTRS1985

Theoretical and software considerations for general dynamic analysis using multilevel substructured models

The dynamic analysis of complex structural systems using the finite element method and multilevel substructured models is presented. The fixed-interface method is selected for substructure reduction because of its efficiency, accuracy, and adaptability to restart and reanalysis. This method is extended to reduction of substructures which are themselves composed of reduced substructures. The implementation and performance of the method in a general purpose software system is emphasized. Solution algorithms consistent with the chosen data structures are presented. It is demonstrated that successful finite element software requires the use of software executives to supplement the algorithmic language. The complexity of the implementation of restart and reanalysis porcedures illustrates the need for executive systems to support the noncomputational aspects of the software. It is shown that significant computational efficiencies can be achieved through proper use of substructuring and reduction technbiques without sacrificing solution accuracy. The restart and reanalysis capabilities and the flexible procedures for multilevel substructured modeling gives economical yet accurate analyses of complex structural systems.

Schmidt, R. J.↗

NASA NTRS1991

Calculating Transient Vibrations Of Coupled Substructures

Finite-element, numerical-integration method for estimating transient vibrational response of structure composed of coupled substructures entails less computation. Responses of substructures to external forces and to forces of interaction between substructures computed. Equations of motion of each substructure solved independently. Applicable to any number and configuration of linearly responding substructures and to both determinate and indeterminate boundary conditions at interfaces between substructures. Also applicable to changing interface boundary conditions.

Admire, J. R.↗

NASA NTRS1993

Substructure system identification and synthesis

This paper explores the possibility of performing system identification at the substructure level and then synthesizing the results to obtain a mathematical model for the assembled structure. The study here shows that in order to enforce interface compatibility and equilibrium conditions to the substructure test data, it is necessary to place collocated actuator/sensor pair at every interface degree-of-freedom. Procedures for assembling substructure transfer function data, substructure state-space models, and substructure Markov parameters are presented. Testing difficulties and possible solutions are also discussed. A numerical simulation example is included to illustrate the proposed substructure synthesis methods.

Su, Tzu-Jeng↗

NASA NTRS1994

A structural design decomposition method utilizing substructuring

A new method of design decomposition for structural analysis and optimization is described. For this method, the structure is divided into substructures where each substructure has its structural response described by a structural-response subproblem, and its structural sizing determined from a structural-sizing subproblem. The structural responses of substructures that have rigid body modes when separated from the remainder of the structure are further decomposed into displacements that have no rigid body components, and a set of rigid body modes. The structural-response subproblems are linked together through forces determined within a structural-sizing coordination subproblem which also determines the magnitude of any rigid body displacements. Structural-sizing subproblems having constraints local to the substructures are linked together through penalty terms that are determined by a structural-sizing coordination subproblem. All the substructure structural-response subproblems are totally decoupled from each other, as are all the substructure structural-sizing subproblems, thus there is significant potential for use of parallel solution methods for these subproblems.

Scotti, Stephen J.↗

NASA NTRS1989

A Transient Response Method for Linear Coupled Substructures

A method is presented for determining the transient response of a discrete coordinate model of a linear structural system composed of substructures. The method is applicable to systems consisting of any number of substructures, both determinate and indeterminate interface boundaries, and any topological arrangement of the substructures. The method is simple to implement from a computational point of view because the equations of motion of each of the substructures are solved independently, and the interface boundary compatibility conditions are enforced at each integration time step by a matrix multiplication. The method is demonstrated for a structural system consisting of two beam segments and acted upon by a time dependent force. The numerical results from the demonstration problem validates the accuracy of the method. The application of this method to structural systems with changing interface boundary conditions between substructures is discussed.

Admire, J. R.↗

NASA NTRS1975

Dynamic substructure analysis of the International Ultraviolet Explorer (IUE) spacecraft

The results are presented of a steady state vibration analysis of the IUE spacecraft simulating its sinusoidal vibration test. The model of the spacecraft, including solar arrays and the scientific instrument, consisted of three separate substructure models one of which was used to represent the two identical solar arrays. Substructuring techniques were used since the large overall size of the problem precluded solving it utilizing a single model. The models used for each substructure are discussed, including reduction to an acceptable size for the combined dynamic analysis. The DMAP alters needed for performing the modal analysis and the subsequent modal frequency response and substructure data recovery are included. Comparison of the results with data obtained during vibration tests of the spacecraft are also included.

Case, W. R.↗

NASA NTRS1977

On the use of attachment modes in substructure coupling for dynamic analysis

Substructure coupling or component-mode synthesis may be employed in the solution of dynamics problems for complex structures. Although numerous substructure-coupling methods have been devised, little attention has been devoted to methods employing attachment modes. In the present paper the various mode sets (normal modes, constraint modes, attachment modes) are defined. A generalized substructure-coupling procedure is described. Those substructure-coupling methods which employ attachment modes are described in detail. One of these methods is shown to lead to results (e.g., system natural frequencies) comparable to or better than those obtained by the Hurty (1965) method.

Craig, R. R., Jr.↗

NASA NTRS1977

Substructure coupling: A different approach

The substructure analysis of the Lockheed L-1011 Dash 500 long range derivative without use of the NASTRAN level 16 multistage substructuring capability is described. It is presented as an example of how a large structural analysis can be organized into manageable independent tasks and how preprocessors, model definition conventions, and data management programs can be used to simplify the data management and model documentation. A procedure is described to analyze a single substructure by modifying the static solution to solve the interface compatibility equations simultaneously with the substructure solution.

Lahey, R. S.↗

NASA NTRS1978

Synthesis and dynamic characteristics of large structures with rotating substructures

The variational equations of motion of large structures with rotating substructures are derived by the substructure synthesis approach, whereby a discretization procedure akin to the Rayleigh-Ritz method is used to represent the elastic motion of every substructure by a suitable set of admissible functions. Using an inclusion principle for gyroscopic systems, the effects on the predicted system dynamic characteristics of truncating the number of admissible functions used for each substructure can be assessed. Criteria for rational selection of admissible functions are presented.

Meirovitch, L.↗

NASA NTRS1983

State vector formulation of substructure coupling for damped systems

A generalized substructure coupling procedure in state vector form is derived for a complex system with general viscous damping. This first-order differential equation formulation is used in order to permit complex substructure modes to be employed easily. The free-interface normal (complex) modes and rigid-body modes of the substructure are defined. Complex residual attachment modes which result from static approximation of neglected higher modes are derived. The motion of each substructure is represented by a selected set of component modes. Equations of interface compatibility are employed to obtain an independent set of system equations of motion. A new method which employs incomplete complex normal modes in conjunction with the complex residual attachment modes to account for the contribution of neglected higher order modes is derived. Examples are employed to indicate how a damped structure may be analyzed by including the effect of residual attachment modes. Numerical results indicate that the new component mode synthesis method provides sytem equations of motion with reduced number of degrees of feedom leading to more accurate approximations to the system frequencies and damping factors than are obtained by pure mode truncation.

Chung, Y.-T.↗

NASA NTRS1984

Parallel triangularization of substructured finite element problems

Much of the computational effort of the finite element process involves the solution of a system of linear equations. The coefficient matrix of this system, known as the global stiffness matrix, is symmetric, positive definite, and generally sparse. An important technique for reducing the time required to solve this system is substructuring or matrix partitioning. Substructuring is based on the idea of dividing a structure into pieces, each of which can then be analyzed relatively indepenently. As a result of this division, each point in the finite element discretization is either interior to a substructure or on a boundary between substructures. Contributions to the global stiffness matrix from connections between boundary points from the K(bb) matrix are reported. The triangularization of a general K(bb) matrix on a parallel machine is specifically discussed.

Leuze, M. R.↗

NASA NTRS1986

A nonlinear substructuring method for concurrent processing computers

This paper proposes a method, based on substructuring, to solve nonlinear structural analysis problems on multiprocessor computers. Background information is given on the use of substructuring in large-scale finite element programs and computational time distributions for the major components for an example nonlinear finite element analysis are discussed. Implementation of the substructuring method on a typical multiprocessor computer is described and estimates are made of expected reductions in computation times based on nonlinear substructuring results obtained on a single processor computer.

Storaasli, O. O.↗