Fumio Fujii

Modified stiffness iteration to pinpoint multiple bifurcation points

Abstract A modified stiffness iteration to precisely compute a bifurcation point with multiple zero eigenvalues is presented. The iteration combines the direct pinpointing iteration for simple bifurcation points with a procedure to modify the tangent stiffness matrix based on pertinent congruent transformation to reduce the multiple eigenvalues to single ones. Two different transformation matrices are employed to modify the stiffness matrix: One is a non-orthogonal transformation matrix amplifying the values of the entries in a certain row and a column of the stiffness matrix. The other is an orthogonal one, which is chosen with reference to the symmetry of the system under consideration. The use of the non-orthogonal transformation matrix is a key idea termed “stiffness amplification method” in this paper, whereas the orthogonal matrix is based on block-diagonalization, which is becoming popular in group-theoretic analysis of symmetric structures. Extensive numerical examples show the robustness and usefulness of the proposed iteration method for pinpointing multiple bifurcation points.

Variable displacement control to overcome turning points of nonlinear elastic frames

Abstract Usual displacement control works well to trace the equilibrium path of nonlinear structures with snap-through. It often breaks down, however, in a snap-back situation, when a turning point of the controlled displacement occurs accidentally on the path. The selection of the controlling parameter is, therefore, of vital importance in displacement control. The present study proposes an intelligent way to overcome the turning points by choosing the current largest component in tangent vector as the best control parameter.

Selection of the control parameter in displacement incrementation

Abstract The conventional displacement incrementation is applicable to simply softening load-deflection curves and snap-through, when a monotonically increasing displacement is controlled. It is, however, believed that the displacement control method will fail when snap-back and looping of the equilibrium path occur. The reason for this limited applicability is that only a fixed nodal parameter is incremented. Therefore, the choice of the control parameter to be specified is important in displacement incrementation. The present paper proposes an automatic determination of the best-suited control parameter in displacement incrementation. The tangent vector of equilibrium path will serve as a monitor of singular points such as limit points and turning points of displacements. The vector component with the maximum absolute value is detected and the corresponding parameter is specified to circumvent the singular points. The control parameter is in this way automatically determined and never fixed. It depends on the local geometry of the solution path and may change in each incremental step. To examine the validity of this variable control parameter method, 2D/3D trusses are computed as benchmark problems. The proposed strategy is generally applicable to nonlinear problems and competitive with the arc-length method.

Finite displacement theory of straight beams under configuration-dependent uniform loads

Abstract A general field transfer matrix for nonlinear elastic curved beams has already been given by Fujii and Gong [J. struct. Engng, ASCE 114, 675–692 (1988)]. For a straight member, this field transfer matrix can be considerably simplified for practical purposes. Firstly, the simplified field transfer matrix is given explicitly. The load terms are also evaluated in explicit form for liquid pressure, wind pressure and gravity-type loading. Secondly, by assuming the element length to be infinitesimally small, the governing differential equations of finite deflection theory are obtained. The nonlinear coefficients in the differential equations are summarized in a 6 × 6 sparse matrix. It is suggested that a higher order field transfer matrix can be derived analytically by utilizing this sparseness. Thirdly by solving the transfer equations for the nodal forces, total stiffness equations are derived for a nonlinear elastic beam element. Finally, an incremental/iterative nonlinear scheme is proposed to solve the total stiffness equations. Numerical examples including looping and snap-back are computed to examine the formulation, and the effects of different uniform loads are also compared.

Computational Stability Theory — Its Strategy

The general stability theory developed by Koiter [1], Thompson/Hunt [2] well applies in the theoretical buckling analysis. The mathematical stability theory seems however not always feasible in computational practice. More specifically, use of the higher-order derivatives of the equilibrium equations, for example, is not realistic in existing finite element codes. The computational stability theory will challenge to stability problems from the more practical and computational viewpoint. The present paper describes the fundamental strategies, path-tracing, pinpointing and path-switching [3,4,5,6,7,8,9], in the computational stability theory.

Buckling Modes of Large-Scale Shell Structures Automatically Detected from Linearized Stiffness by Iterative Solvers

This study presents a novel method to automatically detect bifurcation buckling modes of large scale shell structures during solving a set of linearized stiffness equations in geometrically nonlinear problems by iterative solvers, such as the CG method or the Lanczos method. The proposed method is based on the LDLT decomposition method for direct solvers proposed by the authors [1][2] and is extended for the iterative solvers in order to handle large-scale problems. First, the proposed method detects the approximate buckling mode during the simultaneous process of tri-diagonalization and the LDLT decomposition of the stiffness matrix, utilizing the fact that the Lanczos algorithm still preserves the eigenvalue properties of original matrix during the tri-diagonalization. Second, it is also shown that the correction vector in the direction of solution in the CG method can approximate the buckling mode. The proposed method can avoid a time-consuming eigenanalysis, which is usually necessary for detecting bifurcation modes at critical points, and can compute the approximate bifurcation modes closed to the critical points very efficiently and accurately. Several numerical examples of bifurcation buckling of shells demonstrate the potential of the proposed method

FLEXIBLE AND INCOMPRESSIVE GOAL NETS IN SOCCER

Sports nets and hammocks are usually not pre-tensioned and too slack and flexible to resist compressive forces, which the high pre-stressing would counterbalance in tension structures. Due to this incompressibility, the low-tension nets may exhibit extreme sags in the deformed configuration, which may be one computational problem in analysis. The present study is concerned with this particular, but inter- esting issue and proposes some modelling ideas to simulate the static and dynamic response of flexible and incompressible nets, which are not frequently treated in engineering applications. Special attention is paid to dynamics of hexagon-mesh goal nets used in FIFA World Cup 2002 in Japan and South Korea. In numeri- cal examples, it is illustrated that the hexagon-mesh net is much better than the square-mesh net, especially for more enthusiasm in stadium and also for better visual effects in TV.