Georg Vossen  Torsten Hermanns  Jens Schüttler 《ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik》2015,95(3):297-316

This work introduces a mathematical model for laser cutting taking account of spatially distributed laser radiation. The model involves two coupled nonlinear partial differential equations describing the interacting dynamical behaviors of the free boundaries of the melt during the process. The model will be investigated by linear stability analysis to study the occurence of ripple formations at the cutting surface. We define a measurement for the roughness of the cutting surface and introduce an optimal control problem for minimizing the roughness with respect to the laser beam intensity along the free melt surface. Necessary optimality conditions will be deduced. Finally, a numerical solution will be presented and discussed by means of the necessary conditions. physical considerations.  相似文献   

    

G. Teichelmann  M. Schaub  B. Simeon 《ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik》2005,85(12):864-877

Mathematical models and numerical methods for the computation of both static equilibria and dynamic oscillations of railroad catenaries are derived and analyzed. These cable systems form a complex network of string and beam elements and lead to coupled partial differential equations in space and time where constraints and corresponding Lagrange multipliers express the interaction between carrier, contact wire, and pantograph head. For computing static equlibria, three different algorithms are presented and compared, while the dynamic case is treated by a finite element method in space, combined with stabilized time integration of the resulting differential algebraic system. Simulation examples based on reference data from industry illustrate the potential of such computational tools.  相似文献   

    

K.D. Mombaur  H.G. Bock  J.P. Schlder  R.W. Longman 《ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik》2005,85(7):499-515

We present a numerical method that optimizes the open‐loop stability of solutions of periodic optimal control problems. We consider general periodic processes that may have several phases, each characterized by its own set of differential equations, and discontinuities of the state variables and the right hand side between phases. Stability is measured in terms of the spectral radius of the monodromy matrix which results in a nonsmooth optimization criterion. We have applied this method to design walking robots that can perform stable periodic gaits without any sensors or feedback; three such examples are presented in this paper.  相似文献   

    

Martin Rumpf  Max Wardetzky 《GAMM-Mitteilungen》2014,37(2):184-216

Triggered by the development of new hardware, such as laser range scanners for high resolution acquisition of complex geometric objects, new graphics processors for realtime rendering and animation of extremely detailed geometric structures, and novel rapid prototyping equip‐ment, such as 3D printers, the processing of highly resolved complex geometries has established itself as an important area of both fundamental research and impressive applications. Concepts from image processing have been picked up and carried over to curved surfaces, physically based modeling plays a central role, and aspects of computer aided geometry design have been incorporated. This paper aims at highlighting some of these developments, with a particular focus on methods related to the mechanics of thin elastic surfaces. We provide an overview of different geometric representations ranging from polyhedral surfaces over level sets to subdivision surfaces. Furthermore, with an eye on differential‐geometric concepts underlying continuum mechanics, we discuss fundamental computational tasks, such as surface flows and fairing, surface deformation and matching, physical simulations, as well as spectral and modal methods in geometry processing. Finally, beyond focusing on single shapes, we describe how spaces of shapes can be investigated using concepts from Riemannian geometry. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

    

T.M. Atanackovic  B. Stankovic 《ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik》2007,87(7):537-546

We treat fractional order differential equations that contain left and right Riemann‐Liouville fractional derivatives. Such equations arise as the Euler‐Lagrange equation in variational principles with fractional derivatives. We find solutions of such equations or construct corresponding integral equations.  相似文献   

    

A. C. Marta  C. A. Mader  J. R. R. A. Martins  E. Van der Weide  J. J. Alonso 《International Journal of Computational Fluid Dynamics》2013,27(9-10):307-327

A methodology for the rapid development of adjoint solvers for computational fluid dynamics (CFD) models is presented. The approach relies on the use of automatic differentiation (AD) tools to almost completely automate the process of development of discrete adjoint solvers. This methodology is used to produce the adjoint code for two distinct 3D CFD solvers: a cell-centred Euler solver running in single-block, single-processor mode and a multi-block, multi-processor, vertex-centred, magneto-hydrodynamics (MHD) solver. Instead of differentiating the entire source code of the CFD solvers using AD, we have applied it selectively to produce code that computes the transpose of the flux Jacobian matrix and the other partial derivatives that are necessary to compute sensitivities using an adjoint method. The discrete adjoint equations are then solved using the Portable, Extensible Toolkit for Scientific Computation (PETSc) library. The selective application of AD is the principal idea of this new methodology, which we call the AD adjoint (ADjoint). The ADjoint approach has the advantages that it is applicable to any set of governing equations and objective functions and that it is completely consistent with the gradients that would be computed by exact numerical differentiation of the original discrete solver. Furthermore, the approach does not require hand differentiation, thus avoiding the long development times typically required to develop discrete adjoint solvers for partial differential equations, as well as the errors that result from the necessary approximations used during the differentiation of complex systems of conservation laws. These advantages come at the cost of increased memory requirements for the discrete adjoint solver. However, given the amount of memory that is typically available in parallel computers and the trends toward larger numbers of multi-core processors, this disadvantage is rather small when compared with the very significant advantages that are demonstrated. The sensitivities of drag and lift coefficients with respect to different parameters obtained using the discrete adjoint solvers show excellent agreement with the benchmark results produced by the complex-step and finite-difference methods. Furthermore, the overall performance of the method is shown to be better than most conventional adjoint approaches for both CFD solvers used.  相似文献   

    

N.D. Lopes  P.J.S. Pereira  L. Trabucho 《国际流体数值方法杂志》2012,69(7):1186-1218

An improved class of Boussinesq systems of an arbitrary order using a wave surface elevation and velocity potential formulation is derived. Dissipative effects and wave generation due to a time‐dependent varying seabed are included. Thus, high‐order source functions are considered. For the reduction of the system order and maintenance of some dispersive characteristics of the higher‐order models, an extra O(μ²ⁿ⁺²) term (n ∈ ?) is included in the velocity potential expansion. We introduce a nonlocal continuous/discontinuous Galerkin FEM with inner penalty terms to calculate the numerical solutions of the improved fourth‐order models. The discretization of the spatial variables is made using continuous P₂ Lagrange elements. A predictor‐corrector scheme with an initialization given by an explicit Runge–Kutta method is also used for the time‐variable integration. Moreover, a CFL‐type condition is deduced for the linear problem with a constant bathymetry. To demonstrate the applicability of the model, we considered several test cases. Improved stability is achieved. Copyright © 2011 John Wiley & Sons, Ltd.  相似文献   

    

K. Yamasaki  H. Nagahama 《ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik》2008,88(6):515-520

The J‐integral (a path‐independent energy integral) formalism is the standard method of analyzing nonlinear fracture mechanics. It is shown that the energy density of deformation fields in terms of the homotopy operator corresponds to the J‐integral for dislocation‐disclination fields and gives the force on dislocation‐disclination fields as a physical interpretation. The continuum theory of defects gives a natural framework for understanding the topological aspects of the J‐integral. This geometric interpretation gives that the J‐integral is an alternative expression of the well‐known theorem in differential geometry, i.e., the Gauss‐Bonnet theorem (with genus = 1). The geometrical expression of the J‐integral shows that the Eshelby's energy‐momentum (the physical quantity of the material space) is closely related to the Einstein 3‐form (the geometric objects of the material space).  相似文献   

    

A. Montlaur  S. Fernandez‐Mendez  A. Huerta 《国际流体数值方法杂志》2012,70(5):603-626

The spatial discretization of unsteady incompressible Navier–Stokes equations is stated as a system of differential algebraic equations, corresponding to the conservation of momentum equation plus the constraint due to the incompressibility condition. Asymptotic stability of Runge–Kutta and Rosenbrock methods applied to the solution of the resulting index‐2 differential algebraic equations system is analyzed. A critical comparison of Rosenbrock, semi‐implicit, and fully implicit Runge–Kutta methods is performed in terms of order of convergence and stability. Numerical examples, considering a discontinuous Galerkin formulation with piecewise solenoidal approximation, demonstrate the applicability of the approaches and compare their performance with classical methods for incompressible flows. Copyright © 2011 John Wiley & Sons, Ltd.  相似文献   

    

A.V. Balakrishnan  M.A. Shubov  C.A. Peterson 《ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik》2004,84(5):291-313

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 $mathfrak{S}_p$ 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.  相似文献