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Electrodeposition is a widely used technique for the fabrication of high aspect ratio microstructures. In recent years, much research has been focused within this area aiming to understand the physics behind the filling of high aspect ratio vias and trenches on substrates and in particular how they can be made without the formation of voids in the deposited material. This paper reports on the fundamental work towards the advancement of numerical algorithms that can predict the electrodeposition process in micron scaled features. Two different numerical approaches have been developed, which capture the motion of the deposition interface and 2‐D simulations are presented for both methods under two deposition regimes: those where surface kinetics is governed by Ohm's law and the Butler–Volmer equation, respectively. In the last part of this paper the modelling of acoustic forces and their subsequent impact on the deposition profile through convection is examined. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
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An algorithm, based on the overlapping control volume (OCV) method, for the solution of the steady and unsteady two‐dimensional incompressible Navier–Stokes equations in complex geometry is presented. The primitive variable formulation is solved on a non‐staggered grid arrangement. The problem of pressure–velocity decoupling is circumvented by using momentum interpolation. The accuracy and effectiveness of the method is established by solving five steady state and one unsteady test problems. The numerical solutions obtained using the technique are in good agreement with the analytical and benchmark solutions available in the literature. On uniform grids, the method gives second‐order accuracy for both diffusion‐ and convection‐dominated flows. There is little loss of accuracy on grids that are moderately non‐orthogonal. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
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《ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik》2017,97(1):70-91
In this paper, the governing equations and finite element formulations for a microstructure‐dependent unified beam theory with the von Kármán nonlinearity are developed. The unified beam theory includes the three familiar beam theories (namely, Euler‐Bernoulli beam theory, Timoshenko beam theory, and third‐order Reddy beam theory) as special cases. The unified beam formulation can be used to facilitate the development of general finite element codes for different beam theories. Nonlocal size‐dependent properties are introduced through classical strain gradient theories. The von Kármán nonlinearity which accounts for the coupling between extensional and bending responses in beams with moderately large rotations but small strains is included. Equations for each beam theory can be deduced by setting the values of certain parameters. Newton's iterative scheme is used to solve the resulting nonlinear set of finite element equations. The numerical results show that both the strain gradient theory and the von Kármán nonlinearity have a stiffening effect, and therefore, reduce the displacements. The influence is more prominent in thin beams when compared to thick beams. 相似文献
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In a recent paper a generalized potential flow theory and its application to the solution of the Navier–Stokes equation are developed.1 The purpose of this comment is to show that the analysis presented in that paper is in general not correct. We note that the theoretical development of Reference 1 is in fact an extension—although not cited—of some work first done by Hawthorne for steady inviscid flow.2 Hawthorne's solution is correct, and his analysis, which we briefly describe, provides a useful introduction to this note. 相似文献
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S.A.M. Ghannadpour H.R. Ovesy M. Nassirnia 《ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik》2012,92(8):668-680
This paper presents the theoretical developments of two finite strip methods (i.e. semi‐analytical and full‐analytical) for the post‐buckling analysis of some box section struts. In the semi‐analytical finite strip approach, all the displacements are postulated by the appropriate shape functions while in the development process of the full‐analytical approach, the von‐Kármán's equilibrium equation is solved exactly to obtain the buckling loads and the corresponding form of out‐of‐plane buckling deflection modes. The investigation of struts buckling behaviour is then extended to the post‐buckling study with the assumption that the deflected form after the buckling is the combination of first, second and higher (if required) modes of buckling. Thus, the full‐analytical post‐buckling study is effectively a multi term analysis, which is attempted by utilizing the so‐called semi‐energy method. In this method the von‐Kármán compatibility equation is used together with a consideration of the total strain energy of the strut. Through the solution of the compatibility equation, the in‐plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficients in the assumed out‐of‐plane deflection function. The in‐plane and out‐of‐plane deflection functions are substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficients. It is noted that the Classical Plate Theory (CPT) is applied throughout the theoretical developments. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of these methods is significantly promoted. 相似文献
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V.N. Grebenev M. Oberlack 《ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik》2012,92(3):179-195
Considering the metric tensor ds2(t), associated with the two‐point velocity correlation tensor field (parametrized by the time variable t) in the space 𝒦3of correlation vectors, at the first part of the paper we construct the Lagrangian system (Mt,ds2(t)) in the extended space 𝒦3 × R+ for homogeneous isotropic turbulence. This allows to introduce into the consideration common concept and technics of Lagrangian mechanics for the application in turbulence. Dynamics in time of (Mt,ds2(t)) (a singled out fluid volume equipped with a family of pseudo‐Riemannian metrics) is described in the frame of the geometry generated by the 1‐parameter family of metrics ds2(t) whose components are the correlation functions that evolve according to the von Kármán‐Howarth equation. This is the first step one needs to get in a future analysis the physical realization of the evolution of this volume. It means that we have to construct isometric embedding of the manifold Mt equipped with metric ds2(t) into R3 with the Euclidean metric. In order to specify the correlation functions, at the second part of this paper we study in details an initial‐boundary value problem to the closure model [19,26] for the von Kármán‐Howarth equation in the case of large Reynolds numbers limit. 相似文献
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《ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik》2018,98(9):1666-1685
We study the homogenization of an elastic material reinforced with thin periodic parallel elastic ribbons. We assume perfect adhesion along the common interface between the matrix and the ribbons. We assume that the ribbons are isotropic von Kármán plates of highly contrasting rigidity with respect to that of the matrix. The main result is the derivation of an effective stored energy involving new degrees of freedom implying second and third gradient of displacements. 相似文献
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P. D. Minev 《国际流体数值方法杂志》2008,58(3):307-317
In this paper we demonstrate that some well‐known finite‐difference schemes can be interpreted within the framework of the local discontinuous Galerkin (LDG) methods using the low‐order piecewise solenoidal discrete spaces introduced in (SIAM J. Numer. Anal. 1990; 27 (6): 1466–1485). In particular, it appears that it is possible to derive the well‐known MAC scheme using a first‐order Nédélec approximation on rectangular cells. It has been recently interpreted within the framework of the Raviart–Thomas approximation by Kanschat (Int. J. Numer. Meth. Fluids 2007; published online). The two approximations are algebraically equivalent to the MAC scheme, however, they have to be applied on grids that are staggered on a distance h/2 in each direction. This paper also demonstrates that both discretizations allow for the construction of a divergence‐free basis, which yields a linear system with a ‘biharmonic’ conditioning. Both this paper and Kanschat (Int. J. Numer. Meth. Fluids 2007; published online) demonstrate that the LDG framework can be used to generalize some popular finite‐difference schemes to grids that are not parallel to the coordinate axes or that are unstructured. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
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《ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik》2018,98(8):1412-1419
Possibilities of control and suspension of bending vibrations of a geometrically nonlinear Euler‐Bernoulli beam subjected to two oppositely moving point loads in given finite time are considered. It is assumed that the beam undergoes large deformations and the nonlinear von Kármán strains are considered. The suspension is carried out by means of optimizing the placements of visco‐elastic dampers under the beam. By applying the modified Bubnov‐Galerkin procedure, it becomes possible to avoid linearization of the state equation. The validation of the theory is carried out on the example of a finite simply supported beam. It is observed that by optimizing the dampers placements, both the maximal absolute value of the beam transverse displacements and the vibration reduction time can be reduced. 相似文献
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J.B. Gunda 《ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik》2013,93(8):597-608
Large amplitude free vibration behavior of thin, isotropic square plate configurations resting on Winkler type of elastic foundation are expressed in the form of simple closed‐form solutions by using the Rayleigh‐Ritz (R‐R) method based on coupled displacement fields (CDF). Geometric non‐linearity of von‐Kármán type is taken into consideration. The in‐plane displacement field variations used in the formulation of the R‐R (CDF) method are derived by using the governing in‐plane static differential equations of the plate which in turn simplifies the difficulty of assuming an in‐plane displacement field variations of the plate. Accuracy and robustness of proposed closed‐form solutions is compared to the available finite element formulation results. Proposed closed‐form solutions are corrected for the simple harmonic motion (SHM) assumption using the well established harmonic balance method (HBM) which is applicable for the homogeneous form of cubic non‐linear Duffing equation. 相似文献
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The present study aims to accelerate the convergence to incompressible Navier–Stokes solution. For the sake of computational efficiency, Newton linearization of equations is invoked on non‐staggered grids to shorten the sequence to the final solution of the non‐linear differential system of equations. For the sake of accuracy, the resulting convection–diffusion–reaction finite‐difference equation is solved line‐by‐line using the proposed nodally exact one‐dimensional scheme. The matrix size is reduced and, at the same time, the CPU time is considerably saved due to the decrease of stencil points. The effectiveness of the implemented Newton linearization is demonstrated through computational exercises. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
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An algorithm for a time accurate incompressible Navier–Stokes solver on an unstructured grid is presented. The algorithm uses a second order, three‐point, backward difference formula for the physical time marching. For each time step, a divergence free flow field is obtained based on an artificial compressibility method. An implicit method with a local time step is used to accelerate the convergence for the pseudotime iteration. To validate the code, an unsteady laminar flow over a circular cylinder at a Reynolds number of 200 is calculated. The results are compared with available experimental and numerical data and good agreements are achieved. Using the developed unsteady code, an interaction of a Karman vortex street with an elliptical leading edge is simulated. The incident Karman vortex street is generated by a circular cylinder located upstream. A clustering to the path of the vortices is achieved easily due to flexibility of an unstructured grid. Details of the interaction mechanism are analysed by investigating evolutions of vortices. Characteristics of the interactions are compared for large‐ and small‐scale vortex streets. Different patterns of the interaction are observed for those two vortex streets and the observation is in agreement with experiment. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
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A three-dimensional Navier–Stokes equation is considered. The forcing term is the derivative of a continuous function; the case of white noise is also considered. The aim is to prove the existence of weak solutions and to construct an attractor for the corresponding shift dynamical system in path space, following an idea of Sell. 相似文献
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The non‐reflective boundary conditions (NRBC) for Navier–Stokes equations originally suggested by Poinsot and Lele (J. Comput. Phys. 1992; 101 :104–129) in Cartesian coordinates are extended to generalized coordinates. The characteristic form Navier–Stokes equations in conservative variables are given. In this characteristic‐based method, the NRBC is implicitly coupled with the Navier–Stokes flow solver and are solved simultaneously with the flow solver. The calculations are conducted for a subsonic vortex propagating flow and the steady and unsteady transonic inlet‐diffuser flows. The results indicate that the present method is accurate and robust, and the NRBC are essential for unsteady flow calculations. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
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In this paper we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of laminar flow simulations at low Mach numbers using an implicit scheme. The algorithm is based on the flux preconditioning approach, which modifies only the dissipative terms of the numerical flux. This formulation is quite simple to implement in existing implicit DG codes, it overcomes the time‐stepping restrictions of explicit multistage algorithms, is consistent in time and thus applicable to unsteady flows. The performance of the method is demonstrated by solving the flow around a NACA0012 airfoil and on a flat plate, at different low Mach numbers using various degrees of polynomial approximations. Computations with and without flux preconditioning are performed on different grid topologies to analyze the influence of the spatial discretization on the accuracy of the DG solutions at low Mach numbers. The time accurate solution of unsteady flow is also demonstrated by solving the vortex shedding behind a circular cylinder at the Reynolds number of 100. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献