Hirohisa Noguchi

Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation

In this paper, we establish a homogenization framework to analyze the microscopic symmetric bifurcation buckling of cellular solids subjected to macroscopically uniform compression. To this end, describing the principle of virtual work for infinite periodic materials in the updated Lagrangian form, we build a homogenization theory of finite deformation, which satisfies the principle of material objectivity. Then, we state a postulate that at the onset of microscopic symmetric bifurcation, microscopic velocity becomes spontaneous, yet changing the sign of such spontaneous velocity has no influence on the variation in macroscopic states. By applying this postulate to the homogenization theory, we derive the conditions to be satisfied at the onset of microscopic symmetric bifurcation. The resulting conditions are verified by analyzing numerically the in-plane biaxial buckling of an elastic hexagonal honeycomb. It is thus shown that three kinds of experimentally observed buckling modes of honeycombs i.e., uniaxial, biaxial and flower-like modes, are attained and classified as microscopic symmetric bifurcation. It is also shown that the multiplicity of bifurcation gives rise to the complex cell-patterns in the biaxial and flower-like modes.

Advanced general-purpose computational mechanics system for large-scale analysis and design

Abstract We have been developing an advanced general-purpose computational mechanics system, named ADVENTURE, which is designed to be able to analyze a model of arbitrary shape with a 10–100 million degrees of freedom (DOFs) mesh, and additionally to enable parametric and nonparametric shape optimization. Domain-decomposition-based parallel algorithms are implemented in pre-processes (domain decomposition), main processes (system matrix assembling and solutions) and post-process (visualization), respectively. The hierarchical domain decomposition method with a pre-conditioned iterative solver (HDDM) is adopted in the main processes as one of the major solution techniques. Module-based architecture of the system with standardized I/O format and libraries are also developed and employed to attain flexibility, portability, extensibility and maintainability of the whole system. This paper describes some key technologies employed in the system, and shows some latest results including elastic stress analysis of a precise three-dimensional (3D) model of a nuclear reactor vessel with a 60 million DOF mesh on Hitachi SR2201 (1024 PEs ) , and nonparametric shape optimization of a support structure for an express way with a one million DOF mesh on a PC cluster (10 PEs ) .

Post-buckling analysis of elastic honeycombs subject to in-plane biaxial compression

Abstract In this paper, employing the homogenization theory and the microscopic bifurcation condition established by the authors, we discuss which microscopic buckling mode grows in elastic honeycombs subject to in-plane biaxial compression. First, we focus on equi-biaxial compression, under which uniaxial, biaxial and flower-like modes may develop as a result of triple bifurcation. By forcing each of the three modes to develop, and by comparing the internal energies, we show that the flower-like mode grows steadily if macroscopic strain is controlled, while either the uniaxial or biaxial mode develops if macroscopic stress is controlled. Second, by analyzing several cases other than equi-biaxial compression, it is shown that a second bifurcation from either the uniaxial or biaxial mode to the flower-like mode, which is distorted, occurs under biaxial compression in a certain range of biaxial ratio under macroscopic strain control. Finally, the possibility of macroscopic instability under biaxial compression is discussed.

Elastic–plastic analysis of nuclear structures with millions of DOFs using the hierarchical domain decomposition method

Abstract The hierarchical domain decomposition method (HDDM) proposed by Comp. Sys. Eng. 4 (1993) 495 is applied to the large scale elastic–plastic finite element (FE) analysis of nuclear structures. The HDDM is a method to implement the finite element method (FEM) on various kinds of parallel environments. The substructure-based iterative methods can effectively be used with the HDDM to solve the large scale linear algebraic equations derived from the implicit FEM. In this paper, some key techniques to parallelize the static elastic–plastic FE analysis by the HDDM are described. As illustrative examples, a support structure of the high temperature engineering test reactor (HTTR), a pressure vessel, and an internal pump of a pressure vessel are analyzed. The structure of HTTR and the pressure vessel are modeled by hexahedral solid elements whose total degrees of freedom (DOFs) are about 1.3 millions (M) and 3 M, respectively. The internal pump is modeled by quadratic tetrahedral elements whose total DOFs are about 2 M. The elastic–plastic analysis of a simple cube with 10 M DOFs is also carried out. Both the conjugate gradient method for solving the linear equations and the Newton–Raphson method for solving nonlinear problems successfully converge.

Large scale drop impact analysis of mobile phone using ADVC on Blue Gene/L

Existing commercial finite element analysis (FEA) codes do not exhibit the performance necessary for large scale analysis on parallel computer systems. In this paper, we demonstrate the performance characteristics of a commercial parallel structural analysis code, ADVC, on Blue Gene/L (BG/L). The numerical algorithm of ADVC is described, tuned, and optimized on BG/L, and then a large scale drop impact analysis of a mobile phone is performed. The model of the mobile phone is a nearly-full assembly that includes inner structures. The size of the model we have analyzed has 47 million nodal points and 142 million DOFs. This does not seem exceptionally large, but the dynamic impact analysis of a product model, with the contact condition on the entire surface of the outer case under this size, cannot be handled by other CAE systems. Our analysis is an unprecedented attempt in the electronics industry. It took only half a day, 12.1 hours, for the analysis of about 2.4 milliseconds. The floating point operation performance obtained has been 538 GFLOPS on 4096 node of BG/L.

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.

Recursive bifurcation of tensile steel specimens

Failure modes of steel specimens subjected to uniaxial tension are investigated. These modes are well known to display complex geometrical characteristics of deformation accompanied by the plastic instability behavior. As an underlying mechanism of such complexity, we here focus on the recursive occurrence of bifurcations. In the theory, the rule of recursive bifurcation of a rectangular parallelepiped domain is obtained by the group-theoretic bifurcation theory so as to exhaust all the mathematically possible courses of bifurcation. In the experiment, we examine the representative failure modes with reference to the rules to identify actual courses of recursive bifurcation. Three-dimensional finite element analysis of a thin specimen is conducted to observe the recursive bifurcation, in which diffuse necking is formed by the direct bifurcation and the single shear band by the secondary bifurcation. The recursive bifurcation has thus been identified as the mechanism to create the complex failure modes.

Meshfree analyses of cable-reinforced membrane structures by ALE–EFG method

Abstract Many large membrane structures have been constructed in these days and large membrane structures are often stiffened by cables for reinforcement of the strength. In the analysis of cable-reinforced membrane structures, there are several complicated problems, such as folding of membrane by cable, sliding of cable on membrane surface and so on. As these problems are hardly analyzed by the finite element method, in this study, a meshfree method based on the element free Galerkin (EFG) method is applied to overcome these problems. In the conventional EFG method, the problem that contains discontinuous slope of displacement cannot be analyzed because the moving least square approximation yields C1 continuous displacement field without any modification. Additionally, sliding between cable and membrane surface must be considered. In this study, the arbitrary Lagrangian–Eulerian formulation and simple technique using patches with the penalty method are combined and incorporated into the conventional EFG method. The proposed method is applied to the analyses of cable-reinforced membrane structures to demonstrate the potential of the method. Satisfactory results are obtained and the validity of the method is clarified.

A Polycrystalline Analysis of Hexagonal Metal Based on the Homogenized Method

In this study, a three-dimensional finite element formulation for polycrystalline plasticity model based on the homogenization method has been presented. The homogenization method is one of the useful procedures, which can evaluate the homogenized macroscopic material properties with a periodical microstructure, so-called a unit cell. The present study focuses on hexagonal metals such as titanium or magnesium. An assessment of flow stress by the presented method is conducted and it is clarified how the method can reproduce the behavior of hexagonal metal.

Interface Tracking in Meshfree Methods and its Applications

An enhanced meshfree method, moving least squares approximation with discontinuous derivative basis functions (MLSA-DBF), has been proposed in order to accurately track the derivative discontinuities of continuum or structures. Firstly, quadratic basis functions for MLSA-DBF in three dimensions are presented and the meshfree formulations in Cartesian coordinates are introduced for the analysis of shell structures with slope discontinuities. Numerical examples demonstrate the validity, accuracy, and convergence properties of the proposed method. Secondly, topology optimization with nonlinear materials under large deformation is established based on MLSA-DBF and the level set method. MLSA-DBF can achieve accurate stress and strain fields and obtain accurate sensitivity analysis in the topology optimization problems with fixed/moving material interfaces. The numerical results give faster convergence rate than the method without treatments for material interfaces, and show superior advantages for large deformation problems. It is shown that MLSA-DBF, which is a simple, universal and accurate method without extra parameters, can accurately track not only the material interfaces but also the slope discontinuities and even moving interfaces.