Algebraic Substructuring Research Articles

An algebraic substructuring using multiple shifts for eigenvalue computations

Algebraic substructuring (AS) is a state-of-the-art method in eigenvalue computations, especially for large-sized problems, but originally it was designed to calculate only the smallest eigenvalues. Recently, an updated version of AS has been introduced to calculate the interior eigenvalues over a specified range by using a shift concept that is referred to as the shifted AS. In this work, we propose a combined method of both AS and the shifted AS by using multiple shifts for solving a considerable number of eigensolutions in a large-sized problem, which is an emerging computational issue of noise or vibration analysis in vehicle design. In addition, we investigated the accuracy of the shifted AS by presenting an error criterion. The proposed method has been applied to the FE model of an automobile body. The combined method yielded a higher efficiency without loss of accuracy in comparison to the original AS.

Algebraic substructuring (AS) for eigenvalue and frequency response calculations

AbstractEigenvalue and frequency response calculations are ubiquitous in scientific modeling and engineering analysis. Algebraic substructuring (AS) method is a powerful numerical technique for solving extremely large scale problems. We developed a unified framework and AS code that can solve both problems efficiently. Furthermore, we addressed some of the open problems in this field, including resolving arbitrary eigenmodes, performing high frequency response analysis, accuracy and performance. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An Algebraic Substructuring Method for Large-Scale Eigenvalue Calculation

This paper is concerned with solving large-scale eigenvalue problems by algebraic substructuring. Algebraic substructuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. Through an algebraic analysis, we identify critical conditions under which a simple version of algebraic substructuring works well. This particular version of substructuring is identical to the component mode synthesis (CMS) method (see [R. R. Craig and M. C. C. Bampton, {\em Coupling of substructures for dynamic analysis}, AIAA J., 6 (1968), pp. 1313--1319] and [W. C. Hurty, {\em Vibrations of structure systems by component-mode synthesis}, J. Engrg. Mech., 86 (1960), pp. 51--69]) when the matrix reordering is based on a geometric partitioning of the computational domain. We observe an interesting connection between the accuracy of an approximate eigenpair obtained through substructuring and the distribution of the components of eigenvectors of a canonical matrix pencil congruent to the original problem. A priori error bounds for the smallest eigenpair approximation are developed. This development leads to a simple heuristic for choosing spectral components (modes) from each substructure. The effectiveness of such a heuristic is demonstrated with numerical examples. We show that algebraic substructuring can be effectively used to solve a generalized eigenvalue problem arising from the finite element analysis of an accelerator structure.

Solving large-scale eigenvalue problems in SciDAC applications

Large-scale eigenvalue problems arise in a number of DOE applications. This paper provides an overview of the recent development of eigenvalue computation in the context of two SciDAC applications. We emphasize the importance of Krylov subspace methods, and point out its limitations. We discuss the value of alternative approaches that are more amenable to the use of preconditioners, and report the progress on using the multi-level algebraic sub-structuring techniques to speed up eigenvalue calculation. In addition to methods for linear eigenvalue problems, we also examine new approaches to solving two types of non-linear eigenvalue problems arising from SciDAC applications.

Relative Algebraic Structures

The concept and some of the algebraic properties of the rejective and non-absorptive sets of a subgroup, subring, and subgroup of a module over a ring are investigated. It is shown that the set theoretic complement of a non-absorptive set in the above mentioned algebraic substructures is a normal subgroup (respectively, (left, right) ideal, submodule) of its underlying algebraic structure. The invariant property of the non-absorptive sets under the operation of inversion in the related underlying algebraic structure is proved. $G \setminus R(H)$, the set theoretic complement of the rejective set of a subgroup $H$ in a group $G$, is closed under the product in $G$ and whenever $\vert G \vert$ the order of the group $G$ is finite, $\vert R(H) \vert = (k-s) \vert H \vert$ where each of the $k$ and $s$ is the index of $H$ in $G$ and in $G \setminus R(H)$, respectively. For the case of rings and modules, the set theoretic complement of the rejective set of a substructure in the underlying ring is a subring of the underlying ring. For any subring $S$ of a ring $R$, examples and some of the properties of $S$-relative (left) ideals and $S$-relative submodules are given and also it is shown that $S$ is contained in the set theoretic complement of the rejective set of that $S$-relative (left) ideal (respectively, submodule). Finally, some of the properties of the relative homomorphisms of $R$-modules, and the rejective (respectively, non-absorptive) sets of the group homomorphisms of $R$-modules are investigated.

The join of fuzzy algebraic substructures of a group and their lattices

This is a continuation of work in previous papers [N. Ajmal and K.V. Thomas, Fuzzy Sets and Systems 58 (1993) 217; Inform. Sci. 76 (1994) 1]. Here, we provide a common technique of constructing the join of fuzzy substructures. Consequently, it leads to the formation of various types of lattices and sublattices of fuzzy substructures of a group including those of normal and quasinormal fuzzy subgroups. A lattice theoretic relationship of these sublattices is established. Moreover, we present simple and direct proofs for the fact that the lattice of fuzzy normal subgroups is modular.

Bases of fuzzy algebraic substructures

An axiomatic approach is developed concerning the existence of a basis for certain fuzzy algebraic substructures. Let the set of truth values be a complete Brouwerian lattice L. Let G be an Abelian group. We show that there exists an L-subgroup of G which does not have a p-basis even if G is finite. Let F be a field of characteristic p > 0 and let A, B be L-subfields of F such that A ⊇ B. We show there exists an intermediate L-subfield of A/ B which does not have a relative p-basis over B even if F has finite relative imperfection degree over the support of B. When L = [0, 1], we show that every fuzzy subgroup of G with the sup property has a p-basis and that every intermediate fuzzy subfield of A/ B with the sup property has a relative p-basis if certain compatibility conditions hold.

Finite elements, procedures in engineering analysis

A novel iterative reduced-order modeling method is proposed, which is based on the recently developed algebraic dynamic condensation method. The algebraic substructuring technique is employed to improve the reduction efficiency, and the initial reduced model is calculated using the substructural stiffness condensation and the interface boundary reduction procedures. Then, the initial reduced model is iteratively updated using the iterative substructural inertial effect condensation procedure until the solutions converge. The iterative formulation of the reduced model is represented simply with small submatrix operations to avoid huge computational cost induced by the iterative procedure resulting from the very large global transformation matrix. To verify the performance of the proposed method, we consider several large structural problems, and compare the numerical results to those of the iterated improved reduced system (IIRS) method, a widely used reduced-order modeling method.

Co-hypersimple structures

Priority arguments have been used to produce r.e. sets whose complements have various properties (e.g., maximal sets, hyperhypersimple sets, etc.). In this paper we use the priority method in the context of certain recursively presented models (e.g., dense unbordered linear orderings, countable atomless Boolean algebras, infinite dimensional vector spaces over a finite field) to produce hypersimple subsets of these models whose complements are closed under algebraic operations and/or whose complements are elementarily isomorphic to certain algebraic substructures. For example, it follows from the results of §3 that there are hypersimple subsets of a countable atomless Boolean algebra (of an infinite dimensional vector space over a finite field) in every nonzero r.e. degree whose complement is a subalgebra (subspace). Also, it follows from the same result that, given any subset P of the rationals Q, there is a hypersimple subset of Q whose complement is order isomorphic to P.A model M of a theory T in a countable language L is said to be recursively presented if the universe ∣M∣ of M is an initial segment of the natural numbers and the satisfaction relation “a satisfies Φ in M” is recursive (where a is a finite sequence of elements of ∣M∣ and Φ is a formula in L).

Co-hypersimple structures

Priority arguments have been used to produce r.e. sets whose complements have various properties (e.g., maximal sets, hyperhypersimple sets, etc.). In this paper we use the priority method in the context of certain recursively presented models (e.g., dense unbordered linear orderings, countable atomless Boolean algebras, infinite dimensional vector spaces over a finite field) to produce hypersimple subsets of these models whose complements are closed under algebraic operations and/or whose complements are elementarily isomorphic to certain algebraic substructures. For example, it follows from the results of §3 that there are hypersimple subsets of a countable atomless Boolean algebra (of an infinite dimensional vector space over a finite field) in every nonzero r.e. degree whose complement is a subalgebra (subspace). Also, it follows from the same result that, given any subset P of the rationals Q, there is a hypersimple subset of Q whose complement is order isomorphic to P.A model M of a theory T in a countable language L is said to be recursively presented if the universe ∣M∣ of M is an initial segment of the natural numbers and the satisfaction relation “a satisfies Φ in M” is recursive (where a is a finite sequence of elements of ∣M∣ and Φ is a formula in L).