topology optimization for effective energy propagation in rate independent elastoplastic material systems abstract a topology optimization framework for effective energy management in dynamically loaded structures with rate independent elastoplastic material behavior is presented the elastoplastic constitutive equations are integrated using an implicit numerical scheme based on a multi level newton procedure this primal analysis procedure requires the computation of the incremental algorithmic consistent tangent operator derived from the mappings that enforce the evolving constitutive relations for our gradient based optimization we develop discrete adjoint sensitivity expressions finite element implementation is discussed and the optimization framework is exemplified via the design of three dimensional structures subjected to dynamic loads

algebraic methods for diagonalization of a quaternion matrix in quaternionic quantum theory by means of complex representation and real representation of a quaternion matrix this paper studies the problem of diagonalization of a quaternion matrix gives two algebraic methods for diagonalization of quaternion matrices in quaternionic quantum theory

design sensitivity analysis for rate independent elastoplasticity abstract a new incremental direct differentiation method for design sensitivity analysis of structures with rate independent elastoplastic behavior is presented we formulate analytical sensitivity expressions that are consistent with numerical algorithms for elastoplasticity that use implicit methods to integrate the constitutive equations and return mappings to enforce the consistency conditions the sensitivity expressions can be evaluated with only a modest increase in computational expense beyond the cost of simulation combined with the inherent advantages of implicit integration strategies this represents a significant improvement over previous sensitivity formulations for history dependent materials first order sensitivity expressions involving the complete set of design variables including shape design variables are derived for a generic response functional the reduced form of the consistent tangent stiffness matrix obtained at the end of each time or load step in the finite element procedure is used to update the response sensitivities for that time step no iterations are needed in the sensitivity computations a numerical example demonstrates the accuracy and efficiency of the new sensitivity analysis method for an elastoplastic analysis problem explicit sensitivities from the new method are confirmed by finite difference estimates

indeterminacy in portfolio selection abstract this paper develops the indeterminacy in portfolio selection putting together a modification of a device of measure theory used in our previous papers atti del xxi convegno annuale amases 1997 pp 635 647 soft computing in financial engineering 1999 pp 425 432 and shannon s entropy of information we obtain an expectation variance and indeterminacy e v i functional which is a generalization of the expectation quadratic utility if i 0 our model is more coherent with the expectation variance e v model than the classical model and if i 0 our model yields a warning about the risk from indeterminacy that expectation quadratic utility model does not a numerical method and its statistical application with italian data illustrates the results

dynamic response sensitivity of inelastic structures abstract a general finite element solution method for the dynamic response sensitivity of inelastic structures is developed employing a direct differentiation method the gradient equation of motion is solved without iteration and by taking advantage of the available solution of the response special attention is given to sensitivities with respect to inelastic material parameters and detailed derivations are made for the j 2 plasticity model with a linear hardening rule the method can be applied to any other inelastic material model that has an analytically defined yield function and flow rule the formulation is easily incorporated in existing finite element codes numerical examples demonstrate the accuracy and efficiency of the method

element connectivity parameterization for topology optimization of geometrically nonlinear structures in the current element density based topology optimization element stiffness is penalized to yield either solid or void or very weak materials but low density elements appear during and even after optimization iterations especially when nonlinear problems involving large deformation are considered low density finite elements cause serious numerical problems as their tangent stiffness matrices lose positive definiteness to completely eliminate the problem caused by low density elements we propose a new method all finite elements are kept solid throughout the optimization process zero length elastic links are introduced to parameterize inter element connectivity the link stiffness is penalized in this approach the design variables are defined on the links and vary from 0 and 1 corresponding to the unconnected and rigidly connected states since the finite elements used to discretize the analysis domain always remain solid typical numerical problems encountered by the standard element density based formulation disappear to implement the present method several issues such as the handling of the mass constraint and raster imaging are also investigated

a newton schur alternative to the consistent tangent approach in computational plasticity we introduce a newton schur alternative to the consistent tangent approach in plasticity computations and show that the structure of the consistent tangent approach is preserved in the new newton schur approach however the local constitutive iterations used in the conventional computational plasticity analyses are eliminated when using the newton schur approach two numerical examples are presented which illustrate the efficiencies gained by the new approach

parallel lagrange newton krylov schur methods for pde constrained optimization part i the krylov schur solver large scale optimization of systems governed by partial differential equations pdes is a frontier problem in scientific computation reduced quasi newton sequential quadratic programming sqp methods are state of the art approaches for such problems these methods take full advantage of existing pde solver technology and parallelize well however their algorithmic scalability is questionable for certain problem classes they can be very slow to converge in this two part article we propose a new method for steady state pde constrained optimization based on the idea of using a full space newton solver combined with an approximate reduced space quasi newton sqp preconditioner the basic components of the method are newton solution of the first order optimality conditions that characterize stationarity of the lagrangian function krylov solution of the karush kuhn tucker kkt linear systems arising at each newton iteration using a symmetric quasi minimum residual method preconditioning of the kkt system using an approximate state decision variable decomposition that replaces the forward pde jacobians by their own preconditioners and the decision space schur complement the reduced hessian by a bfgs approximation initialized by a two step stationary method accordingly we term the new method it lagrange newton krylov schur lnks it is fully parallelizable exploits the structure of available parallel algorithms for the pde forward problem and is locally quadratically convergent in part i of this two part article we investigate the effectiveness of the kkt linear system solver we test our method on two optimal control problems in which the state constraints are described by the steady state stokes equations the objective is to minimize dissipation or the deviation from a given velocity field the control variables are the boundary velocities numerical experiments on up to 256 cray t3e processors and on an sgi origin 2000 include scalability and performance assessment of the lnks algorithm and comparisons with reduced sqp for up to 1 000 000 state and 50 000 decision variables in part ii of the article we address globalization and inexactness issues and apply lnks to the optimal control of the steady incompressible navier stokes equations

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

voigt reuss topology optimization for structures with nonlinear material behaviors this work is directed toward optimizing concept designs of structures featuring inelastic material behaviours by using topology optimization in the proposed framework alternative structural designs are described with the aid of spatial distributions of volume fraction design variables throughout a prescribed design domain since two or more materials are permitted to simultaneously occupy local regions of the design domain small strain integration algorithms for general two material mixtures of solids are developed for the voigt isostrain and reuss isostress assumptions and hybrid combinations thereof structural topology optimization problems involving non linear material behaviours are formulated and algorithms for incremental topology design sensitivity analysis dsa of energy type functionals are presented the consistency between the structural topology design formulation and the developed sensitivity analysis algorithms is established on three small structural topology problems separately involving linear elastic materials elastoplastic materials and viscoelastic materials the good performance of the proposed framework is demonstrated by solving two topology optimization problems to maximize the limit strength of elastoplastic structures it is demonstrated through the second example that structures optimized for maximal strength can be significantly different than those optimized for minimal elastic compliance 1997 john wiley sons ltd

symplectic balancing of hamiltonian matrices we discuss the balancing of hamiltonian matrices by structure preserving similarity transformations the method is closely related to balancing nonsymmetric matrices for eigenvalue computations as proposed by osborne j acm 7 1960 pp 338 345 and parlett and reinsch numer math 13 1969 pp 296 304 and implemented in most linear algebra software packages it is shown that isolated eigenvalues can be deflated using similarity transformations with symplectic permutation matrices balancing is then based on equilibrating row and column norms of the hamiltonian matrix using symplectic scaling matrices due to the given structure it is sufficient to deal with the leading half rows and columns of the matrix numerical examples show that the method improves eigenvalue calculations of hamiltonian matrices as well as numerical methods for solving continuous time algebraic riccati equations

filters in topology optimization in this article a modified filtered version of the minimum compliance topology optimization problem is studied the direct dependence of the material properties on its pointwise density is replaced by a regularization of the density field by the mean of a convolution operator in this setting it is possible to establish the existence of solutions moreover convergence of an approximation by means of finite elements can be obtained this is illustrated through some numerical experiments the filtering technique is also shown to cope with two important numerical problems in topology optimization checkerboards and mesh dependent designs copyright 2001 john wiley sons ltd

relative risk value models in this paper we propose a relative risk value model and derive a relative measure of risk for lotteries with positive outcomes under a condition called relative risk independence a decision could be made by explicitly trading off between the relative measure of risk and a measure of value which can either be consistent with some expected utility models or represent nonexpected utility preferences specifically this type of risk value model is associated with power or linear plus power and logarithmic or linear plus logarithmic functions we address some prescriptive and descriptive implications of our relative risk value framework and show that our generalized relative risk value model is very flexible for modeling individuals preferences and can explain many decision paradoxes

real linear quaternionic differential operators the renewed interest in searching for quaternionic deviations of standard complex quantum mechanics resulted in the last years in a better understanding of the quaternionic mathematical tools needed to solve quantum mechanical problems in particular a relevant progress has been achieved in solving eigenvalue problems and differential equations for quaternionic operators the practical methods recently proposed to solve quaternionic and complex linear second order differential equations with constant coefficients represent a fundamental starting point to discuss quaternionic potentials in quantum mechanics and study possible violations from complex theories nevertheless only for a restricted class of real linear quaternionic differential operators namely symmetric operators the solution of differential problems was given in this paper we study real linear quaternionic differential equations the proposed resolution s method is based on the jordan canonical form of real linear quaternionic matrices

the weak and strong stability of algorithms in numerical linear algebra the stability of algorithms in numerical linear algebra is discussed the concept of stability is extended to notions of weak stability and strong stability justifications are given for these extensions and the implications of error analyses in terms of these definitions are discussed the concept of weak stability helps to clarify some of the controversy which has arisen concerning the stability of algorithms for toeplitz systems

a level set method for structural topology optimization this paper presents a new approach to structural topology optimization we represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure furthermore a well founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints the result is a 3d topology optimization technique that demonstrates outstanding flexibility of handling topological changes fidelity of boundary representation and degree of automation we have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods the benefit and the advantages of the proposed method are illustrated with several 2d examples that are widely used in the recent literature of topology optimization especially in the homogenization based methods

topology and shape optimization for elastoplastic structural response abstract it is a common practice to base both material topology optimization as well as a subsequent shape optimization on linear elastic response however in order to obtain a realistic design it might be essential to base the optimization on a more realistic physical behavior i e to consider geometrically or and materially nonlinear effects in the present paper an elastoplastic von mises material model with linear isotropic hardening softening for small strains is used the objective of the design problem is to maximize the structural ductility in the elastoplastic range while the mass in the design space is prescribed with respect to the specific features of either topology or shape optimization for example the number of optimization variables or their local global influence on the structural response different methods are applied for topology optimization problems the gradient of the ductility is determined by the variational adjoint approach in shape optimization the derivatives of the state variables with respect to the optimization variables are evaluated analytically by a variational direct approach the topology optimization problem is solved by an optimality criteria oc method and the shape optimization problem by a mathematical programming mp method in topology optimization a geometrically adaptive procedure is additionally applied in order to increase the efficiency and to avoid artificial stress singularities the procedures are verified by 2d design problems under plane stress conditions

nonlinear structural design sensivitity analysis for path dependent problems part 1 general theory abstract two procedures direct differentiation and adjoint structure for design sensitivity analysis of nonlinear structural systems with path dependent effects are described in this paper continuum formulation is used and both geometric and material nonlinearities are included in the derivations incremental nonlinear analysis based on the total lagrangian concept is adopted the reference volume concept is used to unify the shape and nonshape design problems to show the generality of the theory the derived expressions are reduced to some special cases that have been treated previously in the literature in part 2 of the paper a few analytical example problems will be solved to demonstrate the use of the design sensitivity analysis theory the theory has also beendiscretized elsewhere using finite element approximations for implementation into structural analysis programs these developments offer a firm foundation for design optimization of complex structural systems

a new real structure preserving quaternion qr algorithm abstract new real structure preserving decompositions are introduced to develop fast and robust algorithms for the right eigenproblem of general quaternion matrices under the orthogonally j r s symplectic transformations the francis j r s qr step and the j r s qr algorithm are firstly proposed for j r s symmetric matrices and then applied to calculate the schur form of quaternion matrices a novel quaternion givens matrix is defined and utilized to compute the qr factorization of quaternion hessenberg matrices an implicit double shift quaternion qr algorithm is presented with a technique for automatically choosing shifts and within real operations numerical experiments are provided to demonstrate the efficiency and accuracy of newly proposed algorithms

continuum design sensitivity of transient responses using ritz and mode acceleration methods in this paper a unified continuum based sizing design sensitivity analysis method is developed for the transient dynamic response of built up structures using the direct differentiation method the variational equation of the built up structure is differentiate d with respect to the design variable to obtain a variational design sensitivity equation of the transient response the same finite element analysis model that is used to obtain an approximate solution of the variational equation of the built up structure is also used to obtain an approximate solution of the variational design sensitivity equation for large size built up structures the same superposition method is used both to reduce the dimension of the matrix equation of the built up structure and also to reduce the dimension of the design sensitivity equation the continuum based design sensitivity analysis method can be implemented outside established finite element analysis codes using postprocessing data since it does not require derivatives of the stiffness damping and mass matrices moreover the method is efficient since it does not require derivatives of basis vectors either the ritz or mode acceleration method is used to further improve accuracy and efficiency of both analysis and sensitivity results an example with damping shows that accurate design sensitivity of pointwise transient displacements is obtained using only 10 of basis vectors this example indicates that in the continuum based design sensitivity analysis method the same basis vectors that are used for analysis of the built up structure are suitable for analysis of the design sensitivity equation

topology optimization of continuum structures with local stress constraints we introduce an extension of current technologies for topology optimization of continuum structures which allows for treating local stress criteria we first consider relevant stress criteria for porous composite materials initially by studying the stress states of the so called rank 2 layered materials then on the basis of the theoretical study of the rank 2 microstructures we propose an empirical model that extends the power penalized stiffness model also called simp for solid isotropic microstructure with penalization for inter mediate densities in a second part solution aspects of topology problems are considered to deal with the so called singularity phenomenon of stress constraints in topology design an constraint relaxation of the stress constraints is used we describe the mathematical programming approach that is used to solve the numerical optimization problems and show results for a number of example applications 1998 john wiley sons ltd

lsqr algorithm with structured preconditioner for the least squares problem in quaternionic quantum theory abstract the solution of a linear quaternionic least squares qls problem can be transformed into that of a linear least squares ls problem with jrs symmetric real coefficient matrix which is suitable to be solved by developing structured iterative methods when the coefficient matrix is large and sparse the main aim of this work is to construct a structured preconditioner to accelerate the lsqr convergence the preconditioner is based on structure preserving tridiagonalization to the real counterpart of the coefficient matrix of the normal equation and the incomplete inverse upper lower factorization related to only one symmetric positive definite tridiagonal matrix rather than four so it is reliable and has low storage requirements the performances of the lsqr algorithm with structured preconditioner are demonstrated by numerical experiments

quaternions and special relativity we reformulate special relativity by a quaternionic algebra on reals using real linear quaternions we show that previous difficulties concerning the appropriate transformations on the 3 1 space time may be overcome this implies that a complexified quaternionic version of special relativity is a choice and not a necessity

shape and non shape structural sensitivity analysis for problems with any material and kinematic non linearity abstract the general problem of sizing material selection load parameter and shape sensitivity of non linear structural response is considered the non linear sensitivity path is followed by an incremental strategy employing the continuum based updated lagrangian description the resulting finite element equations are derived both the direct differentiation and adjoint structure strategies are considered the merits of using the consistent tangent matrix for sensitivity calculations are indicated the formulation employs the so called reference volume concept and may be seen as a further attempt to unify the shape and non shape sensitivity analysis procedures

a structure preserving jacobi algorithm for quaternion hermitian eigenvalue problems abstract a new real structure preserving jacobi algorithm is proposed for solving the eigenvalue problem of quaternion hermitian matrix by employing the generalized jrs symplectic jacobi rotations the new quaternion jacobi algorithm can preserve the symmetry and jrs symmetry of the real counterpart of quaternion hermitian matrix moreover the proposed algorithm only includes real operations without dimension expanding and is generally superior to the state of the art algorithm numerical experiments are reported to indicate its efficiency and accuracy

a fast structure preserving method for computing the singular value decomposition of quaternion matrices abstract in this paper we propose a fast structure preserving algorithm for computing the singular value decomposition of quaternion matrices the algorithm is based on the structure preserving bidiagonalization of the real counterpart for quaternion matrices by applying orthogonal jrs symplectic matrices the algorithm is efficient and numerically stable

right eigenvalue equation in quaternionic quantum mechanics we study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n dimensional quaternionic vector spaces for quaternionic linear operators the eigenvalue spectrum consists of n complex values for these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations our discussion is also extended to complex linear operators whose spectrum is characterized by 2n complex eigenvalues we show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products finally we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution

quaternionic eigenvalue problem we discuss the right eigenvalue equation for h c and r linear quaternionic operators the possibility to introduce an isomorphism between these operators and real complex matrices allows us to translate the quaternionic problem into an equivalent real or complex counterpart interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics

multi objective possibilistic model for portfolio selection with transaction cost in this paper we introduce the possibilistic mean value and variance of continuous distribution rather than probability distributions we propose a multi objective portfolio based model and added another entropy objective function to generate a well diversified asset portfolio within optimal asset allocation for quantifying any potential return and risk portfolio liquidity is taken into account and a multi objective non linear programming model for portfolio rebalancing with transaction cost is proposed the models are illustrated with numerical examples

hamilton and jacobi come full circle jacobi algorithms for structured hamiltonian eigenproblems we develop jacobi algorithms for solving the complete eigenproblem for hamiltonian and skew hamiltonian matrices that are also symmetric or skew symmetric based on the direct solution of 4 4 and in one case 8 8 subproblems these structure preserving algorithms produce symplectic orthogonal bases for the invariant subspaces associated with a matrix in any one of the four classes under consideration the key step in the construction of the algorithms is a quaternion characterization of the 4 4 symplectic orthogonal group and the subspaces of 4 4 hamiltonian skew hamiltonian symmetric and skew symmetric matrices in addition to preserving structure these algorithms are inherently parallelizable numerically stable and show asymptotic quadratic convergence

consimilarity of quaternions and coneigenvalues of quaternion matrices first of all by characterizing solutions of the quaternion equation a x x b this paper studies consimilarity of quaternions and some related consequences for the important role of coneigenvalues in consimilarity tranformations of quaternion matrices this paper further derives the relations between principle right coneigenvalues of a quaternion matrix and eigenvalues of the corresponding real representation matrix then based on the real representation matrix an effective algorithm is presented to calculate all coneigenvalues and the associated coneigenvectors of a quaternion matrix finally two numerical examples are given to verify the effectiveness of the proposed algorithm

a structure preserving algorithm for the quaternion cholesky decomposition in this paper we discuss the cholesky decomposition of the hermitian positive definite quaternion matrix for the first time the structure preserving gauss transformation is defined and then a novel structure preserving algorithm which is applied to its real representation matrix is proposed our algorithm needs only real number operations does not depend on the quaternion toolbox for matlab qtfm and has more portability although the flops of our algorithm are theoretically about the same as those based on quaternion arithmetic operations or qtfm numerical experiments show that our algorithm runs faster

interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems abstract the focus of this paper is on interpolation schemes for fictitious domain and topology optimization approaches with structures undergoing large displacements numerical instability in the finite element simulations can often be observed due to excessive distortion in low stiffness regions a new energy interpolation scheme is proposed in order to stabilize the numerical simulations the elastic energy density in the solid and void regions is interpolated using the elastic energy densities for large and small deformation theory respectively the performance of the proposed method is demonstrated for a challenging test geometry as well as for topology optimization of minimum compliance and compliant mechanisms the effect of combining the proposed interpolation scheme with different hyperelastic material models is investigated as well numerical results show that the proposed approach alleviates the problems in the low stiffness regions and for the simulated cases results in stable topology optimization of structures undergoing large displacements

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

structured least squares problems and inverse eigenvalue problems for p q reflexive matrices abstract a new meaningful structured matrix p q reflexive matrix is defined without the common assumption that p or q is unitary a general solution is derived for its structured least squares problem as a necessary and sufficient condition being presented for the solvability of its structured inverse eigenvalue problem structured constrains are firstly given to guarantee the existence of the solution the optimal approximation problem is also considered under spectral constrains

solving simple quaternionic differential equations the renewed interest in investigating quaternionic quantum mechanics in particular tunneling effects and the recent results on quaternionic differential operators motivate the study of resolution methods for quaternionic differential equations in this paper by using the real matrix representation of left right acting quaternionic operators we prove existence and uniqueness for quaternionic initial value problems discuss the reduction of order for quaternionic homogeneous differential equations and extend to the noncommutative case the method of variation of parameters we also show that the standard wronskian cannot uniquely be extended to the quaternionic case nevertheless the absolute value of the complex wronskian admits a noncommutative extension for quaternionic functions of one real variable linear dependence and independence of solutions of homogeneous right h linear differential equations is then related to this new functional our discussion is for simplicity presented for quaternion

matrix lsqr algorithm for structured solutions to quaternionic least squares problem abstract in this paper we employ matrix lsqr algorithm to deal with quaternionic least squares problem in order to find the minimum norm solutions with kinds of special structures and propose a strategy to accelerate convergence rate of the algorithm via right left preconditioning of the coefficient matrices we mainly focus on analyzing the minimum norm hermitian solution and the minimum norm bihermitian solution to the quaternionic least squares problem i j k other structured solutions also can be obtained using the proposed technique a number of numerical experiments are performed to show the efficiency of the preconditioned matrix lsqr algorithm

an algorithm for eigenvalues and eigenvectors of quaternion matrices in quaternionic quantum mechanics by means of complex representation and companion vector this paper studies the problems of eigenvalues and eigenvectors of quaternion matrices and gives a technique of computing the eigenvalues and eigenvectors of the quaternion matrices in quaternionic quantum mechanics

gersgorin type theorems for quaternionic matrices this paper aims to set an account of the left eigenvalue problems for real quaternionic finite matrices in particular we will present the gersgorin type theorems for the left and right eigenvalues of square quaternionic matrices we shall conclude the paper with examples showing and summarizing some differences between complex matrices and quaternionic matrices and right and left eigenvalues of quaternionic matrices

portfolio selection based on fuzzy cross entropy in this paper the kapur cross entropy minimization model for portfolio selection problem is discussed under fuzzy environment which minimizes the divergence of the fuzzy investment return from a priori one first three mathematical models are proposed by defining divergence as cross entropy average return as expected value and risk as variance semivariance and chance of bad outcome respectively in order to solve these models under fuzzy environment a hybrid intelligent algorithm is designed by integrating numerical integration fuzzy simulation and genetic algorithm finally several numerical examples are given to illustrate the modeling idea and the effectiveness of the proposed algorithm

real structure preserving algorithms of householder based transformations for quaternion matrices in this paper we survey three different forms of householder based transformations for quaternion matrices in the literature and propose a new form of quaternion householder based transformation we propose real structure preserving algorithms of these householder based transformations which make the procedure computationally more flexible and efficient we compare the computation counts and assignment numbers of these algorithms we also compare the effectiveness of these real structure preserving algorithms applying to the quaternion qrd and the quaternion svd all these four real structure preserving algorithms are more efficient comparing to the algorithms which apply quaternion toolbox using quaternion arithmetics or algorithms which directly performs real householder transformations on the real representation of a quaternion matrix among these four real structure preserving algorithms the most efficient ones are based on quaternion householder reflection and new proposed householder based transformation

nonlinear structural design sensitivity analysis for path dependent problems part 2 analytical examples abstract in part 1 of this paper a general theory of design sensitivity analysis for nonlinear structures with history dependent effects is given the theory is also reduced to several special cases to show its generality in the present paper three examples are used to analytically verify the theory both the adjoint structure method asm and the direct differentiation method ddm are demonstrated the examples are 1 reaction force sensitivity for an indeterminate beam 2 displacement sensitivity for several cases of an axial force element and 3 stress and displacement sensitivities for a visco elastic problem these examples provide insights for numerical implementation of the theory for more complex design problems

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

topology optimization of energy absorbing structures with maximum damage constraint summary a novel density based topology optimization framework for plastic energy absorbing structural designs with maximum damage constraint is proposed this framework enables topologies to absorb large amount of energy via plastic work before failure occurs to account for the plasticity and damage during the energy absorption a coupled elastoplastic ductile damage model is incorporated with topology optimization appropriate material interpolation schemes are proposed to relax the damage in the low density regions while still ensuring the convergence of newton raphson solution process in the nonlinear finite element analyses an effective method for obtaining path dependent sensitivities of the plastic work and maximum damage via adjoint method is presented and the sensitivities are verified by the central difference method the effectiveness of the proposed methodology is demonstrated through a series of numerical examples copyright 2017 john wiley sons ltd

on left eigenvalues of a quaternionic matrix abstract in this paper we introduce the concept of left and right eigenvalues for a quaternionic matrix and investigate their properties quantities and relationship

quaternionic representations of magnetic groups we study magnetic groups which contain a time inversion operator besides spatial symmetries in the framework of quaternionic group representation theory we obtain a classification of these groups depending on the reducibility of their spatial part and then we cross it with the generalized frobenius schur classification that has been obtained by ourselves elsewhere ten distinct cases arise but only five of them apply to factorizable groups i e groups in which the time inversion operator appears alone and not multiplied by a spatial symmetry we supply examples for these five cases and determine which of them apply to bosonic or fermionic systems respectively finally we discuss the degeneracy of energy levels in the presence of a time reversal symmetry kramers degeneracy in quaternionic quantum mechanics obtaining right values in all cases

a decision model based on expected utility entropy and variance abstract a decision model the expected utility entropy and variance eu ev model is proposed where the decision criterion depends on the uncertainty risk factors entropy and variance and on the subjective expected utility the eu ev model is applied to decision problems existing in the literature and it is shown that they can be well explained with this model

tangent operators and design sensitivity formulations for transient non linear coupled problems with applications to elastoplasticity tangent operators and design sensitivities are derived for transient non linear coupled problems the solution process and the formation of tangent operators are presented in a systematic manner and sensitivities for a generalized response functional are formulated via both the direct differentiation and adjoint methods the derived formulations are suitable for finite element implementations analyses of systems with materials that exhibit history dependent response may be obtained directly by applying the analyses of transient non linear coupled systems rate independent elastoplasticity is investigated as a case study and a problem with an analytical solution is analysed for demonstration purposes

topology optimization of non linear elastic structures and compliant mechanisms abstract a linear elastic response is assumed in most structural topology optimization problems while this assumption is valid for a wide variety of problems it is not valid for structures undergoing large displacements the elastic structural analysis used here accommodates geometric and material non linearities the material density field is filtered to enforce a length scale on the field variation and is penalized to remove less effective intermediate densities the filtering scheme is embedded in the structural analysis to resolve the non existent solution to the solid void topology problem in this way we know precisely what optimization problem is being solved the structural topology optimization formulation is also used to design compliant mechanisms

solutions to matrix equations x axb cy r and x axb cy r abstract the present work proposed an alternative approach to find the closed form solutions of the nonhomogeneous yakubovich matrix equation x a x b c y r based on the derived closed form solution to the nonhomogeneous yakubovich matrix equation the solutions to the nonhomogeneous yakubovich quaternion j conjugate matrix equation x a x b c y r are obtained by the use of the real representation of a quaternion matrix the existing complex representation method requires the coefficient matrix a to be a block diagonal matrix over complex field in contrast in this publication we allow a quaternion matrix of any dimension as an application eigenstructure assignment problem for descriptor linear systems is considered

topology optimization of structures under variable loading using a damage superposition approach we present an original algorithm and accompanying mathematical formulation for topology optimization of structures that can sustain material damage and are subject to multiple load cases with varying configurations damage accumulation is simulated using a coupled non linear brittle damage model the structures are optimized for minimum mass subject to stiffness constraints defined as the compliance evaluated at the end of each loading sequence to achieve robustness of the optimized structures the respective damage fields caused by each individual load case are computed and combined using superposition to simulate a worst case damage field all load cases are then run a second time using the worst case damage distribution as a starting point in this way one effectively accounts for the spectrum of possible load sequences to which the structure may be subjected results from this method are compared with an exhaustive brute force approach in which all non repeating load sequences are analyzed individually for each method the corresponding sensitivities are derived and implemented analytically using a path dependent adjoint method the two approaches are implemented on a series of numerical examples which demonstrate that the superposition method produces structures that are as robust as those obtained using the exhaustive method but require significantly less computational effort copyright 2014 john wiley sons ltd

topology optimization of geometrically nonlinear structures based on an additive hyperelasticity technique abstract this paper presents a simple but effective additive hyperelasticity technique to circumvent numerical difficulties in solving the material density based topology optimization of elastic structures undergoing large displacements by adding a special hyperelastic material to the design domain excessive distortion and numerical instability occurred in the low density or intermediate density elements are thus effectively alleviated during the optimization process the properties of the additional hyperelastic material are established based on a new interpolation scheme which allows the nonlinear mechanical behaviour of the remodelled structure to achieve an acceptable approximation to the original structure in conjunction with the adjoint variable scheme for sensitivity analysis the topology optimization problem is solved by a gradient based mathematical programming algorithm numerical examples are given to demonstrate the effectiveness of the proposed method

variational method for design sensitivity analysis in nonlinear structural mechanics this paper describes a unified variational theory of design sensitivity analysis of linear and nonlinear structures for shape nonshape and material selection problems the concepts of reference volume and adjoint structure are used to develop the unified viewpoint a general formula for design sensitivity analysis is derived several analytical linear and nonlinear examples are used to interpret various terms of the formula and demonstrate its use such analytical examples give insight into the application of the general formula to more complex problems that must be treated numerically adjoint state fields are also interpreted as sensitivities for the functional related to the primary structure

a new structure preserving method for quaternion hermitian eigenvalue problems in this paper we propose a novel structure preserving algorithm for solving the right eigenvalue problem of quaternion hermitian matrices the algorithm is based on the structure preserving tridiagonalization of the real counterpart for quaternion hermitian matrices by applying orthogonal jrs symplectic matrices the algorithm is numerically stable because we use orthogonal transformations the algorithm is very efficient it costs about a quarter arithmetical operations and a quarter to one eighth cpu times comparing with standard general purpose algorithms numerical experiments are provided to demonstrate the efficiency of the structure preserving algorithm

a measure of risk and a decision making model based on expected utility and entropy abstract in this paper we extend the classical decision model under risk to a more general case in which the state of nature corresponding to each risky action may have its own distribution more specifically we propose an expected utility entropy eu e measure of risk and a decision making model based on expected utility and entropy the eu e measure of risk reflects an individual s intuitive attitude toward risk the decision model incorporates the expected utility decision criterion as a special case using this decision model a class of decision problems which cannot be dealt with the expected utility or mean variance criterion reasonably can be solved besides some famous decision paradoxes can be interpreted this decision model can either serve as a descriptive or a normative decision model involving risk

failure mitigation in optimal topology design using a coupled nonlinear continuum damage model abstract this paper presents a novel framework for performing topology optimization of structures while accounting for material damage a quasi static non local brittle damage model is coupled to a linear finite element analysis code for modeling the structural response and the initiation and propagation of damage the optimization problem is parameterized using simp variables and failure mitigation is achieved through the enforcement of constraints on the maximum local damage intensity we derive a computationally efficient adjoint sensitivity formulation that accounts for the path dependence of the coupled analysis problem the algorithm is validated through a series of numerical examples

an adjoint variable method for sensitivity analysis of non linear elastic systems summary an adjoint variable method for design sensitivity analysis of non linear elastic systems is presented the method uses domain parameterization and a mutual form of the hu washizu energy principle and extends results reported in a recent work for linear elastic systems to non linear elasticity non linearities due to finite deformations and non linear hyperelastic constitutive models are considered in contrast to other methods for non linear sensitivity analysis the present formulation can be applied with force displacement or mixed approximate solution methods the mutual energy expression used in the adjoint sensitivity derivation is developed from a non linear extension of the hu washizu energy functional and yields a linear governing equation for the adjoint system this has important ramifications for the computational cost of a sensitivity analyses of non linear systems excluding the cost of determining the response of the system the cost of a sensitivity analysis for a non linear system is essentially the same as that for a linear system finite element implementation of the resulting sensitivity expressions is discussed and two numerical examples are presented the first example involves large deformations of a mooney rivlin body while the second involves design sensitivity analysis for mixed solution methods

a rejoinder on quaternionic projective representations in a series of papers published in this journal a discussion was started on the significance of a new definition of projective representations in quaternionic hilbert spaces in the present paper we give what we believe is a resolution of the semantic differences that had apparently tended to obscure the issues

a numerical method for computing the hamiltonian schur form we derive a new numerical method for computing the hamiltonian schur form of a hamiltonian matrix mathcal m in mathbb r 2n times 2n that has no purely imaginary eigenvalues we demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous time algebraic riccati equations despite the fact that no complete error analysis for the method is yet available the numerical results indicate that if no eigenvalues of mathcal m are close to the imaginary axis then the method computes the exact hamiltonian schur form of a nearby hamiltonian matrix and thus is numerically strongly backward stable the new method is of complexity mathbf o n 3 and hence it solves a long standing open problem in numerical analysis

a numerically stable structure preserving method for computing the eigenvalues of real hamiltonian or symplectic pencils a new method is presented for the numerical computation of the generalized eigenvalues of real hamiltonian or symplectic pencils and matrices the method is numerically backward stable and preserves the structure i e hamiltonian or symplectic in the case of a hamiltonian matrix the method is closely related to the square reduced method of van loan but in contrast to that method which may suffer from a loss of accuracy of order sqrt varepsilon where varepsilon is the machine precision the new method computes the eigenvalues to full possible accuracy

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

special least squares solutions of the quaternion matrix equation a x b with applications in this paper by applying particular structure of the real representations of quaternion matrices and the moore penrose generalized inverse we derive the expressions of the minimal norm least squares solution the pure imaginary least squares solution and the real least squares solution for the quaternion matrix equation a x b the resulting formulas only involve real matrices which are simpler than those reported in yuan et al 2013 the corresponding algorithms only perform real arithmetic which also consider particular structure of the real representations of quaternion matrices therefore are very efficient and easily understood numerical examples are provided to illustrate the efficiency of our algorithms

on the power method for quaternion right eigenvalue problem abstract in this paper we study the power method of the right eigenvalue problem of a quaternion matrix a if a is hermitian we propose the power method that is a direct generalization of that of complex hermitian matrix when a is non hermitian by applying the properties of quaternion right eigenvalues we propose the power method for computing the standard right eigenvalue with the maximum norm and the associated eigenvector we also briefly discuss the inverse power method and shift inverse power method for the both cases the real structure preserving algorithm of the power method in the two cases are also proposed and numerical examples are provided to illustrate the efficiency of the proposed power method and inverse power method