Search

Consider a nondifferentiable complex fractional programming $$ {\rm minimize} \, { {\rm Re} [ f(z, {\bar z}) + (z ^{H}Az) ^{1/2} ] \over {\rm Re} [ g(z, {\bar z}) - (z ^{H}Bz)^{1/2}} \qquad {\rm subject \, to} \, h(z, {\bar z}) \in S \subset C^{m}, \quad z \in C^{m}, \leqno (2) $$ where f, g : C2n → C and h : C2n → Cm are analytic functions, A, B ∈ Cn × n are positive semidefinite Hermitian matrices, S is a polyhedral cone in Cm. In this paper, we establish conditions for the existence of an optimal solution in (P) involving (ℒ, ρ, θ)-quasiconvex/-pseudoconvex analytic functions. Based on the sufficient optimality theorem, we construct a duality model and then establish weak/strong, and strict converse duality theorems. [ABSTRACT FROM AUTHOR]

In 1890, Hess found new special case of integrability of Euler–Poisson equations of motion of a heavy rigid body with a fixed point. In 1892, Nekrasov proved that the solution of the problem of motion of a heavy rigid body with a fixed point under Hess conditions reduces to integrating the second‐order linear differential equation. In this paper, the corresponding linear differential equation is derived and its coefficients are presented in the rational form. Using the Kovacic algorithm, we proved that the Liouvillian solutions of the corresponding second‐order linear differential equation exist only in the case, when the moving rigid body is the Lagrange top, or in the case, when the constant of the area integral is zero. In 1890, Hess found new special case of integrability of Euler–Poisson equations of motion of a heavy rigid body with a fixed point. In 1892, Nekrasov proved that the solution of the problem of motion of a heavy rigid body with a fixed point under Hess conditions reduces to integrating the second‐order linear differential equation. In this paper, the corresponding linear differential equation is derived and its coefficients are presented in the rational form.... [ABSTRACT FROM AUTHOR]

In this paper, the forced nonlinear dynamic behavior of functionally graded material (FGM) plate under external dynamic loads is analyzed by means of a high‐order implicit algorithm. The model is developed using the third‐order shear deformation plate kinematics (TSDT). Unlike previous works, in this contribution, the formulation is written without resorting to any homogenization technique neither rule of mixture nor considering FGM as a laminated composite. To handle integrals in the case of inhomogeneous FGMs, the stress vector is split to four parts and it is written in dimensionless form. The resulting equations of motion are established using the Hamilton principle. Finite element discretization is adopted using a four‐node quadrilateral element with seven degrees of freedom per node. The resolution of the nonlinear equations is made by a high‐order implicit algorithm based on the asymptotic numerical method (ANM) techniques. Numerical comparisons of vibration natural frequencies and dynamic response of FGM plates under external loading with literature and laminate composite modeling results are presented to validate the accuracy and the performance of the proposed model. [ABSTRACT FROM AUTHOR]

We consider a mathematical model of a linear vibrational system described by the second-order differential equation , where M and K are positive definite matrices, called mass, and stiffness, respectively. We consider the case where the damping matrix D is positive semidefinite. The main problem considered in the paper is the construction of an efficient algorithm for calculating an optimal damping. As optimization criterion we use the minimization of the average total energy of the system which is equivalent to the minimization of the trace of the solution of the corresponding Lyapunov equation AX + X A = -I, where A is the matrix obtained from linearizing the second-order differential equation. Finding the optimal D such that the trace of X is minimal is a very demanding problem, caused by the large number of trace calculations, which are required for bigger matrix dimensions. We propose a dimension reduction to accelerate the optimization process. We will present an approximation of the solution of the structured Lyapunov equation and a corresponding error bound for the approximation. Our algorithm for efficient approximation of the optimal damping is based on this approximation. Numerical results illustrate the effectiveness of our approach. The authors consider a mathematical model of a linear vibrational system described by the second-order differential equation , where M and K are positive definite matrices, called mass, and stiffness, respectively. They consider the case where the damping matrix D is positive semidefinite. The main problem considered in the paper is the construction of an efficient algorithm for calculating an optimal damping. As optimization criterion the authors use the minimization of the average total energy of the system which is equivalent to the minimization of the trace of the solution of the corresponding Lyapunov equation AX + X A = -I, where A is the matrix obtained from linearizing the second-order differential equation. Finding the optimal D such that the trace of X is minimal is a very demanding problem, caused by the large number of trace calculations, which are required for bigger matrix dimensions. They propose a dimension reduction to accelerate the optimization process. Then they will present an approximation of the solution of the structured Lyapunov equation and a corresponding error bound for the approximation. This algorithm for efficient approximation of the optimal damping is based on this approximation. Numerical results illustrate the effectiveness of this approach. [ABSTRACT FROM AUTHOR]

Optimal design problems for semiconductor devices with relevant thermal effects can be formulated by help of the energy transport model. In this paper we perform a sensitivity analysis to derive the first-order necessary condition for the optimization. Exploiting the special structure of the KKT system we use a special variant of the classical Gummel iteration to provide a very fast optimization algorithm. Numerical results for a ballistic diode underline the feasibility of our approach. [ABSTRACT FROM AUTHOR]

In this paper some analytical and numerical aspects of time-dependent models with internal variables are discussed. The focus lies on elasto/visco-plastic models of monotone type arising in the theory of inelastic behavior of materials. This class of problems includes the classical models of elasto-plasticity with hardening and viscous models of the Norton-Hoff type. We discuss the existence theory for different models of monotone type, give an overview on spatial regularity results for solutions to such models and illustrate a numerical solution algorithm at an example. Finally, the relation to the energetic formulation for rate-independent processes is explained and temporal regularity results based on different convexity assumptions are presented. [ABSTRACT FROM AUTHOR]

In this paper, a two dimensional hybrid model of a continuously variable transmission with push belt is provided. The system is characterized by a large number of contacts with impacts and friction. These interactions are modeled by uni- and bilateral constraints also considering friction. The established equations of motion are integrated numerically by a time-stepping algorithm. The simulation model allows for the computation of the dynamics of the push belt and all contact forces. [ABSTRACT FROM AUTHOR]

Quaternions are a tool used to describe motions of rigid bodies in ℝ3, (Kuipers, [15]). An interesting application is the topic of moving surfaces (Traversoni, [21]), where quaternion interpolation is used which requires solving equations with quaternion coefficients. In this paper we investigate the well known conjugate gradient algorithm (cg-algorithm) introduced by Hestenes and Stiefel [10] applied to quaternion valued, hermitean, positive definite matrices. We shall show, that the features known from the real case are still valid in the quaternion case. These features are: error propagation, early stopping, cg-algorithm as iterative process with error estimates, applicability to indefinite matrices. We have to present some basic facts about quaternions and about matrices with quaternion entries, in particular, about eigenvalues of such matrices. We also present some numerical examples of quaternion systems solved by the cg-algorithm. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present some recent developments and hints for future researches in control inspired of flatness-based ideas. We focus on motion planning: Steering a system from one state to another. We do not consider stabilization and trajectory tracking. We explain how explicit trajectory parameterization, a property that is central for flat systems, can be useful for the feed-forward control of various oscillatory systems (linear, non-linear , finite and infinite dimensional) of physical and engineering interests. Such parameterization provide simple algorithms to generate in real-time trajectories. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give a sufficient and necessary condition for an acceptable solution of an extended nonlinear complementarity problem. As a consequence, a characterization of an acceptable solution of a vertical implicit nonlinear complementarity problem is also established. [ABSTRACT FROM AUTHOR]

In this paper we discuss a finite element approximation method for solving the Pocklington integro-differential equation for the current induced on a straight, thin wire by an incident harmonic electromagnetic field. We obtain an a posteriori error estimate for finite element approximations of the equation, and we prove the reliability of this estimate. The theoretical results are then used to motivate an adaptive mesh-refining algorithm which generates very efficient meshes and yields optimal convergence rates in numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we study five different types of problems in potential and viscous flows past intersecting circles or spheres in two and three dimensions, respectively, with different angles of intersection. We observe a striking resemblance in the form of the solutions in all these cases by introducing certain operators L and M which generate a group G. By introducing a procedure called ‘closure’ which determines the order of the group G, we give a general method to discuss the problem of flow past two intersecting circles or spheres in potential and Stokes flows with different angles of intersection. [ABSTRACT FROM AUTHOR]

The paper describes and extends to polyatomic gases a general strategy for building a hierarchy of numerical models relating the Boltzmann and the Navier-Stokes equations. It is based on two recent mathematically consistent ansatz, namely gaussian BGK collision models and Levermore's moment expansions, and preserves entropy, hyperbolicity, and relaxation constants. We can then adapt the general adaptive algorithm previously developed for coupling Navier-Stokes equations to local kinetic models. In this process, the Navier-Stokes equations are obtained by a Hilbert asymptotic expansion of the moment's equations, which gives them a kinetic interpretation in terms of a positive distribution function associated to the fifteen moments used in Levermore's expansion. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mortar finite element method for second order elliptic boundary value problems with discontinuous coefficients. At the interface where the coefficient is discontinuous, different triangulations and/or discretizations are coupled by means of Lagrange multipliers. The numerical algorithm is based on the algebraic saddle point formulation. The coupling of P1 conforming finite elements with P1 conforming and nonconforming Crouzeix-Raviart elements as studied. Finally, we present the performance of the adaptive refinement process. [ABSTRACT FROM AUTHOR]

This paper presents a spectral multidomain algorithm based on the influence matrix technique. This algorithm leads to a direct method without any iterative process. We show that domain decomposition allows us to use efficiently spectral approximations for the computation of problems exhibiting a singular solution or a complex geometric configuration. [ABSTRACT FROM AUTHOR]

The optimal shape design of a structure is studied as a material distribution problem. The main goal of this paper is find out the optimal mass distribution for restricted mass of the construction. The analysis is carried out through FEM. The iterative procedure includes two steps. The first one leads to continues distribution of material. In other words the stiffness of the construction flows from very strong to very weak. The second iterative steps leads to discrete 0 or 1 distribution of material. The algorithm solves the problem for the same model boundary during the iterative process. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new class of generalized nonlinear implicit quasivariational inclusions for set-valued mappings. We construct some new iterative algorithms for this generalized nonlinear implicit quasivariational inclusions for set-valued mappings without compactness. We prove the existence of solutions for this generalized nonlinear implicit quasivariational inclusions for set-valued mappings without compactness and the convergence of iterative sequences generated by the algorithms. We also give an application to generalized nonlinear implicit variational inequalities. [ABSTRACT FROM AUTHOR]

The stochastic trapezoidal rule provides the only equidistant discretization scheme from the family of implicit Euler methods (see [12]) which possesses the same asymptotic (stationary) law as underlying continuous time, linear and autonomous stochastic systems with white or coloured noise. This identity holds even when integration time goes to infinity, independent of used integration step size! Especially, the asymptotic behaviour of first two moments of corresponding probability distributions is rigorously examined and compared in this paper. The coincidence of asymptotic moments is shown for autonomous systems with multiplicative (parametric) and additive noise using fixed point principles and the theory of positive operators. The key result turns out to be useful for adequate implementation of stochastic algorithms applied to numerical solution of autonomous stochastic differential equations. In particular, it has practical importance when accurate long time integration is required such as in the process of estimation of Lyapunov exponents or stationary measures for oscillators in mechanical engineering. [ABSTRACT FROM AUTHOR]

This paper presents an improved implicit elastic viscoplastic boundary element approach for a general strain hardening model which includes the mixed strain hardening and, as special cases, both the isotropic and kinematic hardening. An improved implicit scheme related to different yield functions (Tresca, von Mises, Mohr-Coulomb, Drucker-Prager, Modified Zienkiewicz-Pande) is introduced, in which a unified explicit form of the viscoplastic strain derivative matrix H is developed. As compared with the usual implicit scheme, in the improved implicit scheme, the viscoplastic strain rate contains not only the current stress increment but also the viscoplastic strain increment. The improved implicit scheme is combined with two boundary element approaches (pure and mixed BEM). Numerical stability related to the improved schemes is discussed for the time step length limit. Finally, numerical examples, discussion, and comparison with existing research results are presented to illustrate the performance of the improved implicit algorithm. [ABSTRACT FROM AUTHOR]

Copyright of ZAMM -- Journal of Applied Mathematics & Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper we construct a new iterative algorithm for solving a new class of nonlinear variational inequalities with set-valued mapping, and give some convergence analysis of iterative sequences generated by the algorithm. [ABSTRACT FROM AUTHOR]

In previous works we reported the development of symbolic computation tools to automate the design of nonlinear state feedback controllers [1], nonlinear PID controllers [2], nonlinear lag-lead compensators [3], and nonlinear obsevers [4] using the extended linearization method [5]. In this paper we show that a careful analysis of the state variables representation of these classical controllers indicates that all of them are particular cases of the nonlinear extension of a mth order linear filter, consisting of a kth order input derivative operator followed by an output mth order linear dynamical system. Using this two blocks decomposition approach, the design of nonlinear extensions of nth order controllers can be decomposed into two independent subalgorithms: a kth order PD controller algorithm, and a mth order state vector feedback algorithm. Hence, an appropriate, assembly of our symbolic computation tools NL Feedback and NLPID could, in principle, allow to use the extended linearization method to synthesize nonlinear extension of arbitrary nth order linear filter (controllers). [ABSTRACT FROM AUTHOR]

There is growing experimental evidence for non‐affine deformations occurring in different types of fibrous soft tissues; meaning that the fiber orientations do not follow the macroscopic deformation gradient. Suitable mathematical modeling of this phenomenon is an open challenge, which we here tackle in the framework of continuum micromechanics. From a rate‐based analogon of Eshelby's inhomogeneity problem, we derive strain and spin concentration tensors relating macroscopic strain rate tensors applied to the boundaries of a Representative Volume Element (RVE), to strain rates and spins within the tissue microstructure, in particular those associated with fiber rotations due to external mechanical loading. After presenting suitable algorithms for integrating the resulting rate‐type governing equations, a first relevance check of the novel modeling approach is undertaken, by comparison of model results to recent experiments performed on the adventitia layer of rabbit carotid tissue. There is growing experimental evidence for non‐affine deformations occurring in different types of fibrous soft tissues; meaning that the fiber orientations do not follow the macroscopic deformation gradient. Suitable mathematical modeling of this phenomenon is an open challenge, which we here tackle in the framework of continuum micromechanics. From a rate‐based analogon of Eshelby's inhomogeneity problem, we derive strain and spin concentration tensors relating macroscopic strain rate tensors applied to the boundaries of a Representative Volume Element (RVE), to strain rates and spins within the tissue microstructure, in particular those associated with fiber rotations due to external mechanical loading. After presenting suitable algorithms for integrating the resulting rate‐type governing equations, a first relevance check of the novel modeling approach is undertaken, by comparison of model results to recent experiments performed on the adventitia layer of rabbit carotid tissue. [ABSTRACT FROM AUTHOR]

The JKR-adhesive impact of a rigid sphere on a power-law graded half space is studied analytically and numerically under the assumptions of elastic similarity, no-slip and quasi-stationarity. The coefficient of normal restitution is determined analytically. The tangential problem is solved by a numerical algorithm based on the Method of Dimensionality Reduction. The tangential coefficient of restitution is depending on only two properly chosen dimensionless parameters. Thereby the presence of only very weak adhesion severely changes the tangential restitution of the sphere. [ABSTRACT FROM AUTHOR]

For the heat equation in a bounded domain we consider the inverse problem of identifying locations and certain properties of the shapes of small heat-conducting inhomogeneities from dynamic boundary measurements on part of the boundary and for finite interval in time. The key ingredient is an asymptotic method based on appropriate averaging of the partial dynamic boundary measurements. Our approach is expected to lead to very effective computational identification algorithms. [ABSTRACT FROM AUTHOR]

The algorithm for the solution of the first biharmonic problem by the Trefftz method is presented. The solution is purely mathematical and follows Mikhlin [29] approach to the solution of the harmonic problem. As such it fills in the gap in the approximate methods of Mathematical Physics. Validity of the algorithm is demonstrated on two examples. The boundary values of the sought function and its normal derivative must be given explicitly to achieve correct results by the algorithm. For that reason, the presented algorithm is, however, not adequate for the plane problems of the theory of elasticity or for plates with free or simply supported edges. [ABSTRACT FROM AUTHOR]

The method of elastic multibody systems is frequently used to describe the dynamical behavior of the mechanical subsystems in multi-physics simulations. One important issue for the simulation of elastic multibody systems is the error-controlled reduction of the flexible body's degrees of freedom. By the use of second order frequency-weighted Gramian matrix based reduction techniques the distribution of the loads is taken into account a-priori and very accurate models can be obtained within a predefined frequency range and even a-priori error bounds are available. However, the calculation of the frequency-weighted Gramian matrices requires high computational effort. Hence, appropriate approximation schemes have to be used to find the dominant eigenspace of these matrices. In the current contribution, the matrix integral needed for calculating the Gramian matrices is approximated by quadratures using integral kernel snapshots. The number and location of these snapshots have a strong influence on the reduction results. Sophisticated snapshot selection methods based on Greedy algorithms from the reduced basis methods are used to construct the optimal location of snapshot frequencies. The method can be viewed as an automatic determination of optimal frequency weighting and as an adaptive learning of quadrature rules. One ingredient of Greedy algorithms is the need of error measures. To gain computational advantage two different error estimators are derived and used in the Greedy algorithm instead of the absolute or relative error. [ABSTRACT FROM AUTHOR]

The concept of representative directions is intended to generalize one-dimensional material models for uniaxial tension to complete three-dimensional constitutive models for the finite element method. The concept is applicable to any model which is able to describe uniaxial loadings, even to those for inelastic material behavior without knowing the free energy. The typical characteristics of the respected material class are generalized in a remarkable similarity to the input model. The algorithm has already been implemented into the finite element systems ABAQUS and MSC.MARC considering several methods to increase the numerical efficiency. The implementation enables finite element simulations of inhomogeneous stress conditions within technical components, though the input model predicts uniaxial material behavior only. [ABSTRACT FROM AUTHOR]

A primal-dual interior point method for state-constrained parabolic optimal control problems is considered. By a Lavrentiev type regularization, the state constraints are transformed to mixed control-state constraints which, after a simple transformation, can be handled as control constraints. Existence and convergence of the central path are shown. Moreover, the convergence of a short step interior point algorithm is proven in a function space setting. The theoretical properties of the algorithm are confirmed by numerical examples. [ABSTRACT FROM AUTHOR]

We review the classical return algorithm for incremental plasticity in the context of nonlinear programming, and we discuss the algorithmic realization of the SQP method for infinitesimal perfect plasticity. We show that the radial return corresponds to an orthogonal projection onto the convex set of admissible stresses. Inserting this projection into the equilibrium equation results in a semismooth equation which can be solved by a generalized Newton method. Alternatively, an appropriate linearization of the projection is equivalent to the SQP method, which is shown to be more robust as the classical radial return. This is illustrated by a numerical comparison of both methods for a benchmark problem. [ABSTRACT FROM AUTHOR]

The classical problem of buckling of an inextensible elastic column, under the action of a compressive force is examined. The column is made of nonlinearly elastic material for which the stress-strain relation is represented by the Ludwick constitutive law. An approximative formula for determination of the force at immediate post-buckling is given. Further post-buckling solutions are obtained for different values of the nonlinearity parameter by numerical integration using the Runge-Kutta-Fehlberg algorithm, and are presented in non-dimensional diagrams. It is shown that no bifurcation point is found in the case of nonlinearly elastic column. [ABSTRACT FROM AUTHOR]

The family of Newmark and generalized α-methods is extended to constrained mechanical systems by using simultaneous position and velocity stabilization as key ideas. In this way, the acceleration constraints need not be evaluated, and the overall algorithm is about as expensive as the application of a BDF method to the GGL-stabilized equations of motion. Moreover, the RATTLE method of molecular dynamics is included as special case. A convergence analysis of the presented α-RATTLE algorithm shows global second order in both position and velocity variables while the Lagrange multipliers are computed to first order accuracy. Additonally, the property of adjustable numerical dissipation carries over from the unconstrained case. [ABSTRACT FROM AUTHOR]

Numerical solving of real-world problems typically consists of several stages. After a mathematical description of the problem and its proper reformulation and discretisation, the resulting linear algebraic problem has to be solved. We focus on this last stage, and specifically consider numerical stability of iterative methods in matrix computations. In iterative methods, rounding errors have two main effects: They can delay convergence and they can limit the maximal attainable accuracy. It is important to realize that numerical stability analysis is not about derivation of error bounds or estimates. Rather the goal is to find algorithms and their parts that are safe (numerically stable), and to identify algorithms and their parts that are not. Numerical stability analysis demonstrates this important idea, which also guides this contribution. In our survey we first recall the concept of backward stability and discuss its use in numerical stability analysis of iterative methods. Using the backward error approach we then examine the surprising fact that the accuracy of a (final) computed result may be much higher than the accuracy of intermediate computed quantities. We present some examples of rounding error analysis that are fundamental to justify numerically computed results. Our points are illustrated on the Lanczos method, the conjugate gradient (CG) method and the generalised minimal residual (GMRES) method. [ABSTRACT FROM AUTHOR]

In this article size/geometry optimization of trusses is performed using the force method and genetic algorithm. A large number of design variables consisting of cross-sectional areas and nodal coordinates are involved in such an optimization, and due to a large number of constraints, the dimensions of the design space are often numerous and in the case of discrete values for cross sections usually discontinuous. In order to avoid local optima, modified genetic algorithms are developed. Furthermore, the force method is employed to improve the speed of the optimization. In the first phase of the described method, the initial geometry of the truss is fixed and near optimum ranges for the cross-section areas are obtained using the relationships from the force method. In the second phase, the geometry of the structure is altered with the aim of designing lower-weight structures. Within the Genetic Algorithm a new dynamic penalty function is defined and a modified process of reproduction is presented. A contraction process is also employed for the design space using shorter substrings for the design variables. [ABSTRACT FROM AUTHOR]

Many algorithms in the field of nonlinear control theory require Lie derivatives or Lie brackets. Up to now, the computation of these derivatives was practical only for simple systems or for systems with a special structure due to the amount of symbolical computations involved. The authors suggest a new approach which is based on automatic differentiation. This approach circumvents several disadvantages of symbolic computations. Moreover, the methods presented here are applicable not only to systems described by explicitly given mathematical expressions but also to systems given by algorithm using conventional programming languages or dedicated modelling languages, respectively. [ABSTRACT FROM AUTHOR]

Fundamental principles and recent methods for investigating the nonlinear dynamics of legged robot motions with respect to control, stability, and design are discussed. One of them is the still challenging problem of producing dynamically stable gaits. The generation of fast walking or running motions requires methods and algorithms adept at handling the nonlinear dynamical effects and stability issues which arise. Reduced, recursive multibody algorithms, a numerical optimal control method, and new stability and energy performance indices are presented which are well-suited for this purpose. Difficulties and open problems are discussed along with numerical investigations into the proposed gait generation scheme. Our analysis considers both bipedal and quadrupedal gaits. [ABSTRACT FROM AUTHOR]

Copyright of ZAMM -- Journal of Applied Mathematics & Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The test-drive of an automobile along a given test-course can be modeled by formulation of a suitable optimal control problem. For the numerical solution the optimal control problem is discretized by a direct shooting method and transformed into a finite dimensional nonlinear optimization problem. With increasing length of the test-course, the dimension of the nonlinear optimization problem increases as well and its numerical solution becomes very difficult due to stability reasons. Therefore a moving horizon technique with reduced range of vision for the test-driver is introduced. Instead of treating the complete test-course, a comparatively short local sector is considered on which a corresponding local optimal control problem can be solved comfortable. The local solutions are then combined by suitable transient conditions. A numerical example with a realistic car model is given. [ABSTRACT FROM AUTHOR]

An algorithm of Bonnans is generalized to the case of optimal control of a stochastic evolution equation. Properties of the adjoint process are also discussed. [ABSTRACT FROM AUTHOR]

Response surface methods can be applied for building both global and mid-range approximation models, to be used in structural optimization as an interface between analysis tools and optimization algorithms. This approach is particularly promising in those areas, where the structural behaviour or analysis process exhibits noise, because conventional optimization strategies often fail in such situations. This capability, together with the flexible approximation interface between analysis and optimization, results in a robust optimum structural design tool. The method is illustrated by means of a car crash safety problem. [ABSTRACT FROM AUTHOR]

For the prediction of the long term behavior and global analysis of nonlinear dynamical systems the generalized cell mapping is an efficient and powerful numerical tool. The only drawback of this method is the enormous computational effort for higher-dimensional systems. We overcome this problem by several adaptive refinement steps of a very rough starting cell grid, where the adaption is controlled by the long-term dynamics of the system. We illustrate the efficiency by examples. [ABSTRACT FROM AUTHOR]

We generate difference and error formulae of arbitrary consistency order on 3-D unstructured grids for the finite difference method. Therefore we have to collect grid points so that we can determine the influence polynomials on which the generation of the formulae is based. The problem is to select the appropriate points so that we receive a well-structured system of equations of the finite difference method and a good error estimate. We present an algorithm for this selection that is controlled by two parameters. The high quality of the formulae is shown by an example. [ABSTRACT FROM AUTHOR]

Many of well-known algorithms in the context of Computer Aided Geometric Design are based on subdivision techniques. Unfortunately, termination criteria for subdivision mostly require a time-consuming computation of the maximum deviation between any given curve segment and its linear approximation at each subdivision step. We generalize results by Wang for Bézier curves ]3[ and present an approach which in advance specifies the number of necessary subdivision steps to obtain a piecewise linear approximation within an assumed accuracy for a given rational Bézier curve or surface. [ABSTRACT FROM AUTHOR]

A point (x*,λ*) is called a turning point of multiplicity p ≥ 1 of the nonlinear system F(x,δ) = 0, F:ℛ × IR → ℛ if rank δ F(x*, λ*) = n - 1, rank [δ F(x*, λ*)| F (x*, λ*)] = n, and if the Ljapunov-Schmidt reduced function has the normal form g(ζ,μ) = ζ ± μ, g : ℛ × ℛ → ℛ. A minimally extended system F(x,λ) = 0, H(x, λ) = 0 is proposed for defining turning points of multiplicity p, where H : ℛ × ℛ → ℛ is a scalar function which is related to the p-th order partial derivatives of g with respect to ζ. When F depends on m ⩽ p -- 1 additional parameters α ϵ ℛ, the system F(x,λ,α) = 0 can be inflated by m + 1 scalar equations H (x,λ, α) = 0,..., H (x,λ, α) = 0. The functions H, : ℛ × ℛ × ℛ × ℛ depend on certain partial derivatives of g with respect to ζ where H corresponds to H in the case m = 0. The regular solution (x*, λ*, α*) of the extended system of n+m+1 equations delivers the desired turning point (x*, λ*) as first part. For numerically solving these systems, two-stage Newton-type methods are proposed. They require to compute higher order partial derivatives of F with respect to low dimensional subspaces as well as derivatives of implicitly defined related functions. Both tasks are realized via automatic differentiation. The behavior of the algorithms has been tested in case of turning points of multiplicity p = 2, 3. [ABSTRACT FROM AUTHOR]

A computable error estimate is established for the model case of the Poisson-problem. This enables an efficient method in applying the nonconforming Crouzeix-Raviart Elements for adaptive refinement techniques. We study reliability and efficiency of the proposed algorithm both theoretically and numerically. [ABSTRACT FROM AUTHOR]