Showing papers in "Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik in 2004"
TL;DR: In this article size/geometry optimization of trusses is performed using the force method and genetic algorithm and a new dynamic penalty function is defined and a modified process of reproduction is presented.
Abstract: 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.
71 citations
TL;DR: In this article, several numerical methods are applied to solve the Savage-Hutter (SH) equations and compared, including traditional difference schemes, as well as high-resolution NOC (Non-Oscillatory Central Differencing) schemes, in which several second-order TVD (Total Variation Diminishing) limiters and a third-order ENO (Essentially non-oscillation) cell reconstruction scheme are used.
Abstract: The Savage-Hutter (SH) equations of granular avalanche flows are a hyperbolic system of equations determining the distribution of depth and depth-averaged velocity components tangential to the sliding bed. We review the equations and point out the geometrical complexities to which these equations have been generalized. Because of the hyperbolicity of the equations, successful numerical modelling is challenging, particularly when large gradients of the physical variables occur, e.g. for a moving front or possibly formed shock waves in avalanche flows if velocities change from supercritical to subcritical e.g. during the deposition. Numerical schemes solving these free surface flows must be able to cope with smooth as well as non-smooth solutions. In this paper several numerical methods are applied to solve the SH equations and compared, including traditional difference schemes, e.g. central and upstream difference schemes, as well as high-resolution NOC (Non-Oscillatory Central Differencing) schemes, in which several second-order TVD (Total Variation Diminishing) limiters and a third-order ENO (Essentially Non-Oscillatory) cell reconstruction scheme are used. Results show that the high-resolution schemes, particularly the NOC scheme with the Minmod TVD limiter or the van Leer limiter, provide excellent performances. In the SH theory the material response is expressed by only two phenomenological parameters - the internal and the bed friction angles. Parameter investigations show that avalanche flows are much more sensitive against variations of the bed friction angle than that of the internal angle of friction. Effects due to a pressure dependence of the bed friction angle and lateral variations of the basal topography are therefore also numerically examined.
68 citations
TL;DR: A new index reduction technique is discussed for the treatment of differential‐algebraic systems for which extra structural information is available and instead of using expensive subspace computations the index reduction is obtained by introducing new variables.
Abstract: In this paper a new index reduction technique is discussed for the treatment of differential-algebraic systems for which extra structural information is available. Based on this information reduced derivative arrays are formed and instead of using expensive subspace computations the index reduction is obtained by introducing new variables. The new approach is demonstrated for several important classes of differential-algebraic systems, where the structural information is available. These include multi-body systems and circuit simulation problems. The effectiveness of the new approach is demonstrated via numerical examples.
66 citations
TL;DR: A general analytic framework is developed which allows an elegant representation of first and second order derivatives of the objective functional and of the state equations which guarantees local quadratic convergence of Newton's method.
Abstract: Second order methods for open loop optimal boundary control problems governed by the instationary Navier-Stokes system are investigated. A general analytic framework is developed which allows an elegant representation of first and second order derivatives of the objective functional and of the state equations. Moreover a second order sufficient optimality condition is proved which guarantees local quadratic convergence of Newton's method.
56 citations
TL;DR: An overview over old and new arguments in the proof of reliability and efficiency of the error estimator η𝒜 := ‖ph ‐ Aph‖ as an approximation of theerror ‖p ‐ ph‖ in (an energy norm) ‖⋅‖ is given.
Abstract: Given a flux or stress approximation ph from a low-order finite element simulation of an elliptic boundary value problem, averaging or (gradient-)recovery techniques aim the computation of an improved approximation Aph by a (simple) post processing of ph. For instance, frequently named after Zienkiewicz and Zhu, Aph is the elementwise interpolation of the nodal values (Aph)(z) obtained as the integral mean of ph on a neighbourhood of z. This paper gives an overview over old
56 citations
TL;DR: In this article, one-dimensional wave propagation through two layers consisting, respectively, of an elastic and a viscoelastic medium is considered, where the material coefficients change smoothly without a discontinuity near the interface.
Abstract: We consider one-dimensional wave propagation through two layers consisting, respectively, of an elastic and a viscoelastic medium The viscoelastic medium is of Kelvin-Voigt type We assume that the material coefficients change smoothly without a discontinuity near the interface It is proved that the decay rate of high frequency modes tends to infinity with frequency
52 citations
TL;DR: An exact solution to the moving boundary problem with fractional anomalous diffusion (in time) in drug release devices is presented by means of fractional Green's function and Wright function in this article.
Abstract: An exact solution to the moving boundary problem with fractional anomalous diffusion (in time) in drug release devices is presented by means of fractional Green's function and Wright function in this paper. The result given by this paper coincides with the known semi-empirical Ritger-Peppas' formula in controlled drug release system [4]. For convenience of use, an approximate expression of the amount of drug released from polymeric matrix with high accuracy is given out at the end of the paper.
36 citations
TL;DR: In this article, a fractal cohesive law is defined to describe the tensile failure of a heterogeneous material, which is scale invariant, and new mathematical operators from fractional calculus are proposed to handle the fractal quantities.
Abstract: Fractal patterns often arise in the failure process of materials with a disordered microstructure. It is shown that they are responsible of the size effects on the parameters characterizing the material behaviour in tensile tests (i.e. the strength, the fracture energy, and the critical displacement). Based on fractal geometry, a simple model of a generic disordered material is set. The physical quantities describing the stress-strain state of such fractal medium are pointed out. They show anomalous (non integer) physical dimensions. In terms of these fractal quantities, it is possible to define a fractal cohesive law, i.e. a constitutive law describing the tensile failure of an heterogeneous material, which is scale invariant. Then we propose new mathematical operators from fractional calculus to handle the fractal quantities previously introduced. In this way, the static and kinematic (fractional) differential equations of the model are pointed out. These equations form the basis of the mechanics of fractal media. In this framework, the principle of virtual work is also obtained.
34 citations
TL;DR: In this article, the use of random sets of probability measures is demonstrated in the practical context of an integrated slope hydrology and stability model called CHASM, which generates a lower and upper cumulative probability distribution on the slope Factor of Safety.
Abstract: When the variables (x 1 ,..., x n+q ) of a function y = g(x 1 ,..., x n+q ) are described by a joint probability distribution over variables x 1 ,...,x n and a random relation over variables x n+1 ,...,x n+q , the dependent variable y is described by a random set of probability measures. Starting with a brief review of the theory of random sets, the formalism of a random set of probability measures is introduced. One approach to constructing a random set from a set of interval estimates from different sources is reviewed. The use of random sets of probability measures is demonstrated in the practical context of an integrated slope hydrology and stability model called CHASM. When the parameters defining the slope stability problem are described by a random set of probability measures the analysis using CHASM generates a lower and upper cumulative probability distribution on the slope Factor of Safety. It is demonstrated how point measurements can be used to update prior imprecise information on geotechnical parameters.
28 citations
TL;DR: In this paper, the authors show how recent generalizations of the usual calculus of probability can be utilized to deal powerfully with complex uncertainty in decision problems, and also a general applicable algorithm is proposed to calculate optimal decision functions by linear programming.
Abstract: A powerful application of decision theory to engineering problems often has failed: The uncertainty underlying is too complex to be modelled adequately by a (precise) probability distribution. The present paper shows how recent generalizations of the usual calculus of probability can be utilized to deal powerfully with complex uncertainty in decision problems. Basic notions of the resulting theory of generalized expected loss and generalized risk are developed and discussed. In addition to this, also a general applicable algorithm is proposed to calculate optimal decision functions by linear programming.
28 citations
TL;DR: In this article, the existence of global in time regular solutions of the Navier-Stokes equations in a domain Ω C R 3 with vanishing Dirichlet boundary conditions was proved.
Abstract: We prove the existence of global in time regular solutions of the compressible Navier-Stokes equations in a domain Ω C R 3 with vanishing Dirichlet boundary conditions. The solutions are close to nontrivial static solutions. A key element of the proof is a special L p -estimate for the linearized problem to show that the velocity belongs to W 2,1 r(loc) (Ω × [0, ∞)) with r > 3 which is the sharp result in this approach.
TL;DR: In this paper, Girding's derivation in the scalar case was extended to derive the Green's tensor for real homogeneous hyperbolic systems of partial differential operators.
Abstract: Modifying L. Girding's derivation in the scalar case we deduce Herglotz-Petrovsky formulae for fundamental matrices (Green's tensors) for real homogeneous hyperbolic systems of partial differential operators. As an application, we calculate the fundamental matrix for elastodynamic systems of hexagonal symmetry with reducible determinant (Props. 1, 2). A special case thereof is the fundamental matrix of the system of uniaxial optics (Prop. 3). The calculations are based on integrals of the type ∫ C ([ξ,p]/[ξ,q]) δ([ξ, x])do(ξ) where the 1904. conic section C in R 4 is defined as the intersection of [ξ, ξ] = 0 with [ξ,N] = 1.
TL;DR: In this paper, an analytic solution of the flow of a second-grade fluid on a porous plate is constructed, where the porous plate executes oscillations in its own plane with superimposed injection or suction.
Abstract: An analytic solution of the flow of a second-grade fluid on a porous plate is constructed. The porous plate is executing oscillations in its own plane with superimposed injection or suction. An increasing or decreasing velocity amplitude of the oscillating porous plate is also examined. It is also shown that in case of second-grade fluid, a combination of suction/injection and decreasing/increasing velocity amplitude is possible as well. Several limiting situations with their implications are given and discussed.
TL;DR: In this paper, exact closed-formed solutions using elliptic integrals for large deflection analysis of elastica of a beam with variable arc-length subjected to an inclined follower force are presented.
Abstract: This paper presents exact closed-formed solutions using elliptic integrals for large deflection analysis of elastica of a beam with variable arc-length subjected to an inclined follower force. The beam is hinged at end but slides freely over the support at the other end. In the undeformed state, the inclined follower force applied at any distance from the hinged end making an angle γ with respect to vertical axis while in the deformed state its direction remains at an angle γ from the normal to the beam axis. The set of nonlinear equations is obtained from the boundary conditions, and solved iteratively for the solutions. The effect of the direction and the position of the follower force on the beam bending behaviour is demonstrated. Comparisons of equilibrium configurations of the beam under non-follower force and follower force are also given.
TL;DR: In this article, a new and simple bound for the exponential decay of second-order systems using the spectral shift is presented, which is applied to finite matrices as well as to partial differential equations of Mathematical Physics.
Abstract: We present a new and simple bound for the exponential decay of second order systems using the spectral shift. This result is applied to finite matrices as well as to partial differential equations of Mathematical Physics. The type of the generated semigroup is shown to be bounded by the upper real part of the numerical range of the underlying quadratic operator pencil.
TL;DR: In this article, the authors considered periodic feedback control systems of differential equations with delays and established a criterion for the existence of a positive periodic solution for each of the problems with delays.
Abstract: This paper considers periodic feedback control systems of differential equations with delays. Applying a continuation theorem, a criterion for the existence of a positive periodic solution is established.
TL;DR: In this article, a stochastic convex optimization problem with complete fixed recourse (SLP) is presented. But the SLP is not applicable to the case of linearized yield/strength conditions.
Abstract: The basic mechanical conditions in optimal plastic design are the convex, linear or linearized yield/strength condition and the linear equilibrium equation for the generic stress (state) vector. Moreover, based on the mechanical survival conditions, the failure costs may be represented by the minimum value of a convex and often linear program. The basic optimal plastic design problem must be replaced by an appropriate deterministic substitute problem, if stochastic variations of the model parameters (e.g. yield stresses, plastic capacities) and external loadings have to be taken into account. Then the total expected costs are minimized subject to the remaining deterministic constraints. A stochastic convex optimization problem is obtained. A “Stochastic Linear Program (SLP) with complete fixed recourse” results when working with linearized yield/strength conditions. In case of a discretely distributed probability distribution or after the discretization of a more general probability distribution of the random structural parameters and loadings as well as certain random cost factors one has a linear program (LP) with a so-called “dual decomposition data” structure. Some numerical examples are considered to demonstrate the solution procedures.
TL;DR: The paper is on statistical analysis for fuzzy data, fuzzy densities and its application in Bayesian inference, including a gereralized probability concept, called fuzzy probability distributions.
Abstract: There are different kinds of uncertainty in connection with statistics and probability models. Besides the fuzziness of data also the imprecision of a-priori distributions is an essential feature of stochastic models in applied mathematical work. The paper is on statistical analysis for fuzzy data, fuzzy densities and its application in Bayesian inference, including a gereralized probability concept, called fuzzy probability distributions.
TL;DR: The formal description of data uncertainty as fuzzy randomness combines randomness and fuzziness in an uncertain model that permits the simultaneous consideration of randomness, fuzziness and fuzzyrandomness and yields the fuzzy stochastic finite element method.
Abstract: The formal description of data uncertainty as fuzzy randomness combines randomness and fuzziness in an uncertain model. This generalized uncertainty model permits the simultaneous consideration of randomness, fuzziness and fuzzy randomness. Variables and functions with the property of fuzzy randomness are introduced. The formulation of the latter using α-levels and fuzzy bunch parameters permits the application of fuzzy randomness in the numerical simulation of engineering problems. The finite element method is extended by the inclusion of fuzzy randomness to yield the fuzzy stochastic finite element method (FSFEM). Examples demonstrate the advantage of the chosen formal description in practical applications.
TL;DR: In this article, the damping properties of viscoelastic media can be described by fractional derivatives under certain assumptions on the interactions between macromolecules, which leads to a macroscopic stress strain relation with fractional derivative in damping term.
Abstract: In good coincidence with experimental data the damping properties of viscoelastic media can be described by fractional derivatives. We present a model, which under certain assumptions on the interactions between macromolecules leads to a macroscopic stress strain relation with fractional derivatives in the damping term. For this purpose we use the Kubo formula of nonequilibrium thermodynamics.
TL;DR: In this article, a time-marching study of non-steady flow around a rotating and oscillating circular cylinder of a viscous incompressible micropolar fluid, for low values of Keulegan-Carpenter number K c and different values of Stokes parameter β is being undertaken.
Abstract: The 'time-marching study' of non-steady flow around a rotating and oscillating circular cylinder of a viscous incompressible micropolar fluid, for low values of Keulegan-Carpenter number K c and different values of Stokes parameter β is being undertaken. In the present studies we have investigated the Magnus effects on stirring or orbital flow for different values of rotation parameter a in the range 0 to 4.5 at various time levels. In this numerical attempt we have adopted the scheme, which consists of two steps. In the first step a 4th order special finite-difference method is used to approximate the constitutive equations. This method transforms the governing partial differential equations to a system of finite-difference equations, which are solved numerically by S.O.R. iterative method. In the second step, the results obtained are further refined and upgraded by Richardson's extrapolation method. That is why this scheme yields the sixth order accurate solution. To check the accuracy of the results these results are compared on five different grid sizes. The results compare very well.
TL;DR: In this paper, a computer assisted method for computing eigenvalue enclosures for non-selfadjoint problems with Blasius profile is presented. But the method is limited to the case of the Orr-Sommerfeld equation.
Abstract: The Orr-Sommerfeld equation is one of the governing equations of hydrodynamic stability. Mathematically, it constitutes a non-selfadjoint eigenvalue problem. Depending on its spectrum being contained in the right complex half-plane or not, the underlying flow is stable or unstable under some given perturbation. Here, we focus on the Blasius profile modelling a flow along a wall. We present a computer-assisted method for computing eigenvalue enclosures for such non-selfadjoint problems. As a specific result, for a particular parameter constellation in the Orr-Sommerfeld equation (often used as test example in the engineering literature), we enclose an eigenvalue in a circle which is completely contained in the left half-plane. This constitutes the first rigorous proof of instability for the Orr-Sommerfeld equation with Blasius profile.
TL;DR: The design of an extended ESP-controller and its consequences for severe braking conditions during cornering are discussed showing the necessary steps in modelling and their mathematical formulation.
Abstract: A general problem formulation and the resulting scheme of modelling dynamical system behaviour is introduced before a specification with respect to automobiles is done. An essential component for the vehicle behaviour is the tyre. Models of different complexity with respect to its mathematical-mechanical formulation are presented. Simple models of the automobile itself allow a separation of longitudinal, lateral, and vertical dynamics and a partially analytical derivation of corresponding equations of motion. These models are usually also the approximations used for the design of a state observer and controller to influence the vehicle behaviour. For a more detailed representation of the vehicle 3D-models are assembled by their individual components, e.g. suspensions, drive train, and steering system. In this way a good agreement between measurements and simulation results can be realized up to the nonlinear range of vehicle behaviour. Nowadays MBS-models, established with professional software packages, are applied. They integrate all essential components, taking into account different linkages, e.g. bushings and joints. For component strength design, a rough road surface description can be implemented additionally. To improve by simulation the active safety or handling of the vehicle, these models represent the controlled complex vehicle in the global control loop whereas linearized models as mentioned above are applied for the controller design. As an example the design of an extended ESP-controller and its consequences for severe braking conditions during cornering are discussed showing the necessary steps in modelling and their mathematical formulation.
TL;DR: In this paper, an approach to compute bounds for the structural reliability by imprecise parameters of the stress and strength probability distributions is proposed, which is based on using imprecize probability theory and takes into account different types of independence of the stressed, strength and their parameters.
Abstract: An approach to compute bounds for the structural reliability by imprecise parameters of the stress and strength probability distributions is proposed. The approach is based on using imprecise probability theory and takes into account different types of independence of the stress, strength and their parameters. It is shown that computation of the imprecise stress-strength model can be reduced to solution of a number of linear programming problems. Special cases of the exponentially distributed stress and strength are considered. Various numerical examples illustrate the approach and show the impact of the independence conditions on imprecision of results.
TL;DR: In this paper, a phenomenological model describing stress and temperature induced transformations in polycrystalline shape memory alloys is proposed, which is mimicked on the level of a finite element discretization.
Abstract: We propose a phenomenological model describing stress and temperature induced transformations in polycrystalline shape memory alloys. Polycrystallinity is mimicked on the level of a finite element discretization: Each element is treated as a single grain of a randomly chosen orientation. This heterogeneity destroys the undesirable effect of instantaneous transformation under spatially homogeneous loading or heating. We present various computational experiments on the hysteretic effects.
TL;DR: A comprehensive application of fuzzy set theory in structural engineering is presented, including non‐stochastic uncertainty is quantified using fuzzy values, and α‐level optimization combined with a modified evolution strategy is formulated.
Abstract: In this paper a comprehensive application of fuzzy set theory in structural engineering is presented. Non-stochastic uncertainty is quantified using fuzzy values. The fuzzy structural parameters are processed on the basis of a generally applicable numerical method for arbitrary linear and nonlinear fuzzy structural analyses. This method is formulated in terms of α-level optimization combined with a modified evolution strategy. The fuzzy structural responses are compared with permissible values and assessed using an analog to the Shannon entropy and defuzzification algorithms. Referring to permissible fuzzy structural responses uncertain structural design parameters are derived by applying a fuzzy cluster analysis to the fuzzy structural parameters. The nonlinear fuzzy structural analysis including uncertain structural design is demonstrated by way of an example.
TL;DR: In this paper, the Strouhal number S d is related to geometrical quantities through S d = C. (d/w) n with n 1, in contrast to some analytical treatments of the problem.
Abstract: We study both, by experimental and numerical means the fluid dynamical phenomenon of edge tones. Of particular interest is the verification of scaling laws relating the frequency f to given quantities, namely d, the height of the jet, w, the standoff distance and the velocity of the jet. We conclude that the Strouhal number S d is related to the geometrical quantities through S d = C . (d/w) n with n 1, in contrast to some analytical treatments of the problem. The constant C of the experiment agrees within 13-15% with the result of the numerical treatment. Only a weak dependence on the Reynolds number with respect to d is observed. In general, a very good agreement of the experimental and the numerical simulations is found.
TL;DR: In this paper, the spectral analysis of a system of coupled Euler-Bernoulli and Timoshenko beams has been studied and it has been shown that the dynamics generator of the system is a Riesz spectral operator in the sense of Dunford.
Abstract: The present paper is devoted to the asymptotic and spectral analysis of a system of coupled Euler-Bernoulli and Timoshenko beams. The model is governed by a system of two coupled differential equations and a two parameter family of boundary conditions modelling the action of the self-straining actuators. The above equations of motion form a coupled linear hyperbolic system, which is equivalent to a single operator evolution equation in the energy space. That equation defines a semigroup of bounded operators. This is a dynamics generator of the semigroup which is our main object of interest in the present paper. We prove that for each set of boundary parameters, the dynamics generator has a compact inverse and this inverse operator belongs to class O p of compact operators with p > 1. We also show that if both boundary parameters are not purely imaginary numbers, then the dynamics generator is a nonselfadjoint operator in the energy space. However, its inverse operator is a finite-rank perturbation of a selfadjoint operator. The latter fact is crucial for the proof of the fact that the root vectors of the dynamics generator form a complete and minimal set in the energy space. We will use the spectral results in our forthcoming papers to prove that the dynamics generator of the system is a Riesz spectral operator in the sense of Dunford and to use the latter fact for the solution of several boundary and distributed controllability problems via the spectral decomposition method.
TL;DR: The authors suggest a new approach which is based on automatic differentiation which circumvents several disadvantages of symbolic computations and is applicable not only to systems described by explicitly given mathematical expressions but also to systems given by algorithms using conventional programming languages or dedicated modelling languages, respectively.
Abstract: 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 algorithms using conventional programming languages or dedicated modelling languages, respectively.