Multi-level Substructuring Research Articles

Buckling and Vibration Analysis of Laminated Panels Using VICONOPT

The analysis aspects of the 23,000‐line FORTRAN program VICONOPT are described. Overall stiffness matrices assembled from the earlier exact VIPASA flat plate stiffnesses are optionally coupled by Lagrangian multipliers to find critical buckling loads, or natural frequencies of undamped vibration, of prismatic assemblies of anisotropic flat plates with arbitrarily located point supports or simple transverse supporting frames. The longitudinal continuity of typical wing and fuselage panels is closely approximated because the solutions are for the infinitely long structure obtained by repeating a bay and its supports longitudinally. Any longitudinally invariant in‐plane plate stresses are permitted, and very rapid solutions are guaranteed by numerous refinements, including multilevel substructuring and a method for repetitive cross sections that is exact for regular polygons used to represent cylinders. Modal displacements and stresses in or between plies of laminated plates are calculated and plotted, with values being recovered at all nodes of substructures. Comparison with usual approximate finite‐element methods confirms that, for comparably converged solutions, VICONOPT is typically between 100 and 104 times faster.

Adaptive multi-level substructuring for computing acoustic radiation and scattering from complex structures

An adaptive method for computing acoustic radiation and scattering from complex structures is presented. The method uses a multi-level substructuring decomposition of a structure’s finite element model, in which the structure is partitioned into several substructures, and each of these is partitioned into its own substructures, and so on. For each substructure, the finite element representation is transformed to a new one in terms of a quasistatic dependence of internal degrees of freedom on interface and excited ones, complemented with response in substructure fixed-interface modes of vibration. This allows for model reduction in the form of modal truncation for substructures on various levels. The order of the model is reduced to the number of degrees of freedom excited by either the acoustic fluid or applied excitations, plus the number of modes included. An adaptive procedure is used to determine which substructure modes should be included in the structure model. The acoustic fluid is represented with a boundary element model, and the procedure for coupling the structure and fluid models is described. The method is demonstrated on numerical examples in three dimensions. [Work supported by ONR.]

Analytical solutions for axisymmetric radiation and scattering from a spherical shell with internal circular plates

Analytical solutions for problems of radiation and scattering from a spherical thin shell containing one or more circular thin plates are presented. The geometry of both the structure and the loading is axisymmetric. These solutions can be used to evaluate computational methods for radiation and scattering from finite three-dimensional structures having internal complexity. In this case the internal complexity is associated with the internal plates, which are represented in terms of their axisymmetric modes of vibration. The motion of the spherical shell is represented in terms of spherical harmonics. The analytical solutions are verified with numerical results obtained using an adaptive multi-level substructuring approach, which obtains an optimally reduced finite element model of the structure, which is coupled with a boundary element model of the acoustic fluid. [Work supported by ONR.]

Review of exact buckling and frequency calculations with optional multi-level substructuring

This review covers the many applications of the Wittrick-Williams algorithm, which ensures that no critical buckling loads, or natural frequencies of undamped free vibration, are missed even when using the ‘exact’ member equations obtained by solving the appropriate differential equations. The review includes: plane and space frames; prismatic assemblies of isotropic or anisotropic plates, including in-plane plate shear loads; exact multi-level substructuring; design; damping; efficient solution of rotationally or linearly repetitive structures; use of Lagrangian multipliers; programmable pocket calculator methods; program listings for small computers and; references to large computer programs.

Multi-level substructuring and its implementation in programming

With the development of the finite element method, FEM, increasing complex structures are being studied and the number of simultaneous equations to be processed is considerable. For many such complex structures, CPU-time, memory storage and data preparation can be reduced by means of multi-level substructuring. Multi-level substructuring and its implementation in computer software are the subject of this paper. With the program developed by authors, any-level of substructuring problem can be conveniently solved.

OPTIMUM DESIGN OF LAMINATED COMPOSITE STRUCTURES VIA AMULTILEVEL SUBSTRUCTURING APPROACH

This paper presents a multilevel substructuring and optimization approach to the minimum weight design of laminated composite structures. The optimization process is carried out in a double scheme which consists of optimizations at system and subsystem levels. At the system level of optimization, an optimality criterion method is used to design component thicknesses which minimize structural weight subject to structural behavioral constraints as well as side constraints. At the subsystem level, the structure being divided into several substructures, fiber directions and layer thicknesses of each substructure are determined to minimize its weight subject to component behavioral constraints as well as side constraints. The objective at the subsystem level is accomplished by carrying out the minimization process again in a double scheme where the quasi-Newton method is used at the first sub-level of optimization for the optimal design of fiber directions and an optimality criterion method at the second sub-l...

Dynamic superelement substructure method in forced vibration analysis

A new mode superposition substructure method with dynamic superelements is presented for forced vibration analysis of multilevel substructure systems. The algorithms derived in this paper are further development on the studies of dynamic superelement method for free vibration analysis by the authors. Thus a compatible method for both free vibration and forced vibration analyses of substructure systems with multilevel superelements has been implemented. In this method, one can take full advantage of a simple static substructure method and eliminate the errors resulting from static condensations. It has no restriction on substructuring. The formulations are frequency independent and not involved with nonlinear eigen problems. Therefore, the method is adapted to arbitrary multilevel substructure systems for free vibration and forced vibration analyses with both simplicity and accuracy.

Hierarchical poly tree configurations for the solution of dynamically refined finite element models

It is demonstrated how a multilevel substructuring technique, called the Hierarchical Poly Tree (HPT), can be used to integrate a localized mesh refinement into the original finite element model more efficiently. The optimal HPT configurations for solving isoparametrically square h-, p- and hp-extensions on single and multiprocessor computers is derived. In addition, the reduced number of stiffness matrix elements that must be stored when employing this type of solution strategy is quantified. Moreover, the HPT inherently provides localized ‘error-trapping’ and a logical, efficient means with which to isolate physically anomalous and analytically singular behavior.

Buckling and vibration of long plate structures by complex finite strip methods

Finite strip methods are presented for the prediction of buckling stresses and natural frequencies of vibration of “long” prismatic plate structures which may be formed of fibre-reinforced, composite, laminated material with very general properties. The finite strip methods are of the single-term type with complex algebra employed to accommodate applied in-plane shear stress and anisotropic material behaviour. The developments, described here follow on very directly from an earlier paper in this Journal ([1] Dawe and Craig, Int. J. Mech. Sci. 30, 77, 1988) to which frequent reference is made herein. The first development is the introduction of major improvements and extensions on the earlier work [1] which is based upon the use of first-order shear deformation plate theory to represent the out-of-plane properties of plate flats: the chief advance involves the use of multi-level substructuring procedures, including the introduction of so-called superstrips, but eccentric connections of component plates at their junctions is also included. The second development is the introduction of a new finite strip analysis which is based on the use of classical plate theory and which is complementary to the shear deformation analysis, with similar advanced features. Two computer programs, BAVPAS and BAVPAC, are introduced and description is given of some results of the application of these programs.

A buckling analysis capability for use in the design of composite prismatic plate structures

The prediction of the buckling stresses of prismatic plate structures made of composite laminated material is considered. Recent developments which have provided an extended capability in the finite strip analysis of such structures are brought together and discussed. Different analysis procedures are described, dependent upon whether the plate structure is of finite length, with diaphragm ends, or is ‘long’, and on whether first-order shear deformation plate theory or classical plate theory is used in developing out-of-plane strip properties. Features of the analysis include a very general description of laminate material properties, the presence of applied shear stress, the accommodation of eccentric connections between plate flats and the use of multi-level substructuring techniques, including the so-called superstrip concept. The associated computer software is used to examine the buckling behaviour of two types of panel, with attention paid to the influence of through-thickness shear deformation and of bending-stretching material coupling in unsymmetric laminates.

Buckling and vibration of finite-length composite prismatic plate structures with diaphragm ends, part I: finite strip formulation

A finite strip approach is presented for the determination of buckling stresses and natural frequencies of vibration of prismatic plate structures of finite length which have diaphragm ends. The structures are assembled from plate flats which generally are laminates of fibre-reinforced composite material and a very broad specification of material properties is allowed which includes anisotropy and bending/stretching coupling. Modes of buckling or vibration may be of an overall or a local type. To accommodate the presence of applied in-plane shear stress (as well as biaxial direct stress) and material anisotropy the finite strip method is of the multi-term type, i.e., in the strip formulation each displacement-type component is represented by a finite series of products of longitudinal trigonometric functions and crosswise polynomial functions. The out-of-plane properties of plate flats are based in turn on the use of first-order shear deformation plate theory (accounting for the through-thickness shear effects which are often significant) and classical plate theory. Strip properties, as represented by the elastic stiffness matrix, the geometric stiffness matrix and the consistent mass matrix, are derived through potential energy principles and are approximate. However, through the use of a repetitive substructuring scheme, so-called superstrips are created economically with properties which are effectively exact within the confines of the particular background theory. Following a transformation to a global coordinate system, incorporating the effects of eccentric connections as well as accounting for the arbitrary inclination of plate flats, multi-level substructuring procedures are invoked to reduce the number of effective freedoms of the structure. The eigenvalues are determined using the extended Sturm sequence-bisection approach. This paper contains no numerical applications: consideration of such applications is deferred to a companion paper.

Multi-level substructuring and an experimental self-adaptive newton-raphson method for two-dimensional nonlinear analysis

The substructuring technique has been employed in structural analysis for 25 years. Until now, the use of this technique in nonlinear analysis has been limited to localized situations. Moreover, the benefit in calculation time one can obtain is restricted to the amount of 50%. However, as will be described in this paper, a new scheme of multi-level substructuring technique in nonlinear analysis has been developed. With this scheme an efficiency of 75–85% has been reached and a connection level higher than three can easily be expanded. The Newton-Raphson method and its modified version are the most well known algorithms in nonlinear numerical analysis. The former has, nevertheless, the severe disadvantage of reforming the stiffness matrix at every iteration and, in fact, some of the stages are not necessary. The latter has also the disadvantage of a slow convergence, as the recalculation of the stiffness matrix is executed in certain fixed iterations. Thus, to overcome the difficulties mentioned above, an experimental self-adaptive Newton-Raphson algorithm, which offers an automatic decision as to whether the stiffness matrix is reformed or not, is introduced. In applying this algorithm a saving of 30–50% in execution time can be achieved. Combining these two new schemes in 2-D nonlinear analysis, a dramatic effort of 85–90% is yielded. A typical flow chart and two numerical examples are also presented.

Multi-level substructuring in the elasto-plastic domain

The application of the multi-level substructuring technique to the analysis of elasto-plastic and local nonlinear problems is studied. After a review of principles and formulations a practical implementation and a typical flow chart of the procedure for the technique are presented. Two numerical examples are provided to demonstrate the efficiency that can be reached by using a multi-level substructuring algorithm.

Subspace iteration method in multi-level substructure systems

An accurate and efficient method for solving eigen-problems of arbitrary multi-level substructure systems is presented in this paper. The basic idea of this method is to use the subspace iteration method combined with the tree travelling technique in a substructure system to obtain the eigen-pairs of the overall structure. General formulas are derived without any approximation and restriction on substructuring.

CONCENTRATED LOADS ON SOLID MASONRY WALLS - A PARAMETRIC STUDY AND DES IGN RECOMMENDATIONS.

The strength of solid masonary walls subjected to concentrated loads is studied, using a finite element model to simulate wall behaviour. The finite element model is able to reproduce the non-linear deformation characteristics of the bricks and mortar as well as local failure in both bricks and joints. To allow the efficient modelling of storey height walls, multi-level substructuring and mesh-refinement techniques are used. A parametric study is carried out with the influence of loaded area ratio, load location and wall geometry being considered. It is shown that all three of these variables significantly influence the degree of enhancement of wall bearing strength in the region directly beneath the load. Existing design rules are reviewed and shown to be non-conservative and incomplete in many cases. More realistic design provisions are prepared from the results of the parametric study. Two relationships are presented, one incorporating all of the above three variables, and a second simplified expression based on a conservative assumption regarding wall geometry.(a)

Multilevel substructuring sensitivity analysis

Solution techniques for handling large scale engineering optimization problems are reviewed. Potentials for practical applications as well as their limited capabilities are discussed. A new solution algorithm for design sensitivity is proposed. The algorithm is based upon the multilevel substructuring concept to be coupled with the adjoint method of sensitivity analysis. There are no approximations involved in the present algorithm except the usual approximations introduced due to the discretization of the finite element model. It is a well-known fact that substructuring concept can be exploited in order to reduce computer core memory requirement and the total solution time executed. These advantages have been realized not only for statics analysis, but also for design sensitivity analysis. For truly large scale structures, however, it is necessary to use multilevel substructuring. The present paper has formulated a multilevel algorithm for design sensitivity analysis. A two-level substructuring example is used to verify the proposed algorithm. The sensitivity vector obtained by the multilevel algorithm is exactly identical to the one obtained by a more conventional approach.

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.

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.

Concepts of a General Substructuring System for Structural Dynamics Analyses

Multilevel substructuring algorithms in structural dynamics are considered together with their software implementation. The paper emphasizes a consistent data structure for representing objects on hierarchical levels. The data structure permits recursive numerical procedures to be easily implemented. As an example, a recursive multi-subspace iteration algorithm is introduced. The paper also considers the definition and implementation of a problem specification language. The language is similar to and retains the expressive power of Lisp. Finally, a numerical example is given.

A computer adaptive language for the development of structural analysis programs

A group of FORTRAN 77 subroutines is presented which are designed to augment the standard fortran language and to produce a new higher-order, machine-independent language for the development of structural engineering software. The group of subroutines which comprise the Computer Adaptive Language for the development of Structural Analysis Programs, CAL/SAP, is designed to operate effectively on micro, supermini and mainframe computers. The CAL/SAP system has been used as the basis for the development of the SAP-80 series of programs and for CAL-80, a series of interactive programs for Computer Assisted Learning of structural analysis and design. The subroutines which define the CAL/SAP development system are divided into the following three categories: • • First, a series of free-field input routines allows input data to be specified in a consistent manner, in arbitrary order, with optional name identification and in arithmetic statement form. • • Second, a set of incore data management subroutines allows dynamic storage allocation to be accomplished with integer, real and ASCII data with a minimum of programming effort. These subroutines eliminate paging problems on modern super minicomputers with virtual operating systems. • • Third, an out-of-core data management system allows different programs to access the same data. Simple operations allow data transfer between the in and out-of-core systems. The out-of-core data base provides for sequential, direct access and bulk data files. Communication between different data bases allows techniques such as multilevel substructure analysis to be implemented with a minimum of programming. The use of the CAL/SAP development system allows computer programs to be rapidly developed and maintained. Also, it can be used to upgrade existing software in order to obtain modularity and to operate efficiently on the new generation of computer systems. The purpose of this paper is to present the CAL/SAP development system and to illustrate that computer independent programs in structural engineering and computational mechanics can be developed which operate on both large and small computers.