Three-dimensional Partial Differential Equations Research Articles

The drag of a freely sendimentating sphere in a sheared weakly viscoelastic fluid

We study the steady, creeping viscoelastic flow around a freely rotating rigid sphere in a stream with uniform velocity which is subject to simple shear flow far from the sphere. The sphere moves along the vorticity axis of the shear flow. The viscoelasticity of the stream is modelled using the Oldroyd-B and Upper Convected Maxwell constitutive equations. A spherical coordinate system with origin at the centre of the sphere is used to describe the flow field. The solution of the governing equations is expanded as a series for small values of the Deborah number and the resulting sequence of three-dimensional partial differential equations is solved analytically up to second order in the Deborah number. The solution is used to deduce the resulting drag force on the sphere in terms of the dimensionless parameters of the flow. The expression for the drag shows that there is a significant increase of the drag due to the shearing, except in the Newtonian case.

Drug Release Kinetics from Biodegradable Polymers via Partial Differential Equations Models

In order to achieve prescribed drug release kinetics some authors have been investigating bi-phasic and possibly multi-phasic releases from blends of biodegradable polymers. Recently, experimental data for the release of paclitaxel have been published by Lao et al. (Lao and Venkatraman in J. Control. Release 130:9---14, 2008; Lao et al. in Eur. J. Pharm. Biopharm. 70:796---803, 2008). In Blanchet et al. (SIAM J. Appl. Math. 71(6):2269---2286, 2011) we validated a two-parameter quadratic ordinary differential equation (ODE) model against their experimental data from three representative neat polymers. In this paper we provide a gradient flow interpretation of the ODE model. A three-dimensional partial differential equation (PDE) model for the drug release in their experimental set up is introduced and its parameters are related to the ones of the ODE model. The gradient flow interpretation is extended to the study of the asymptotic concentrations that are solutions of the PDE model to determine the range of parameters that are suitable to simulate complete or partial drug release.

Heat-Modeling of Microwave Assisted Epoxidation of Palm Acid Oil

Problem statement: While microwave chemistry could well be the most convenient, rapid and energy-saving way to initiate a chemical reaction, the issues of thermal versus microwave specific effect remain unaddressed. Approach: A three-dimensional Finite Element Model (FEM) was developed to predict temperature profile during the enzymatic epoxidation of palm acid oil using microwave heating. Three-dimensional partial differential equations are formulated for the electromagnetic field and heat and mass transfer processes. The temperature profile of the FEM generated from the commercial software (ANSYS Emag) was verified using experimental data from microwave oven heated samples containing palm acid oil and reactants in cylindrical bottles and has compared favorably with real time data. An Infra Red thermometer was used to measure temperature of the reactants during the real time experiment to ensure less contact with the reaction vessel. Results: Apart from ascertaining the temperature profile of the epoxidation process in the cylindrical bottle, the heat-modeling of this reaction can predict the Energy required during the reaction and provide insights to non-thermal microwave effects. It has shown the possibility of Microwave Non-Thermal Effects and Microwave Selectivity in Substrate Specificity. The energy used by the microwave irradiated reaction is by far much lesser than the energy used in conventional heating. This study has also improved the epoxidation process of Palm Acid Oil by incorporating enzyme as a catalyst and using microwave heating. Epoxidation can be done rapidly and at a lower level of activation energy of 27.767 kJ mol−1. It is a green process with increased efficiency as it reduced a typically 5-hour reaction to less than an hour. Conclusion: In this study, a quantitative view of a microwave assisted process of enzymatic epoxidation has been established. We have successfully developed and experimentally validated a 3D electromagnetic heat transfer model. This model can thus be used to predict the energy of the epoxidation and other future microwave reactions. the study also have shown the existence of microwave non-thermal effects and microwave selectivity in substrate specificity.

Meshless global radial point collocation method for three-dimensional partial differential equations

This study deals with the numerical solution of three-dimensional partial differential equations by the meshless global radial point collocation method based on various radial basis functions. First, second, third, and fourth-order three-dimensional partial differential equations are considered. The effect of shape parameters of various radial basis functions on the numerical accuracy is studied. The effect of grid pattern on accuracy is also studied by several numerical examples.

Alternating Direction Finite Volume Element Methods for Three-Dimensional Parabolic Equations

This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The L2 norm and H1 semi-norm error estimates are obtained for the first scheme and second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.

A Representation of solutions to a three-dimensional nonmodel singular partial differential equation

A Representation of solutions to a three-dimensional nonmodel singular partial differential equation

Variational-splitting time integration of the multi-configuration time-dependent Hartree-Fock equations in electron dynamics

We discuss the numerical approximation of the solution to the multi-configuration time-dependent Hartree-Fock (MCTDHF) equations in quantum dynamics. The MCTDHF method approximates the high-dimensional wave function of the time-dependent electronic Schrodinger equation by an antisymmetric linear combination of products of functions depending only on three-dimensional spatial coordinates. The equations of motion, obtained via the Dirac–Frenkel time-dependent variational principle, consist of a coupled system of three-dimensional nonlinear partial differential equations and ordinary differential equations. We investigate the convergence properties of a time integrator based on a splitting of the Hamiltonian directly in the variational principle. First-order convergence in the H1 Sobolev norm and second-order convergence in the L2 norm are established under a solution regularity of H2. As a prerequisite, we show that the MCTDHF equations have a solution in this Sobolev space if the initial data has such regularity.

A new coupled approach high accuracy numerical method for the solution of 3D non-linear biharmonic equations

In this paper, we derive a new fourth order finite difference approximation based on arithmetic average discretization for the solution of three-dimensional non-linear biharmonic partial differential equations on a 19-point compact stencil using coupled approach. The numerical solutions of unknown variable u ( x , y , z ) and its Laplacian ∇ 2 u are obtained at each internal grid point. The resulting stencil algorithm is presented which can be used to solve many physical problems. The proposed method allows us to use the Dirichlet boundary conditions directly and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. The new method is tested on three problems and the results are compared with the corresponding second order approximation, which we also discuss using coupled approach.

An efficient domain decomposition method for three-dimensional parabolic problems

A non-overlapping domain decomposition algorithm to solve three-dimensional parabolic partial differential equations is presented. It has been shown in this paper that the algorithm is unconditionally stable and efficient. Spectral radii for the interface and interior region are provided. Unlike two-dimensional problem, it has been found out that estimating the values of the points of the interface in three-dimensional problem is no longer negligible.

Matrix decomposition RBF algorithm for solving 3D elliptic problems

In this study, we propose an efficient algorithm for the evaluation of the particular solutions of three-dimensional inhomogeneous elliptic partial differential equations using radial basis functions. The collocation points are placed on concentric spheres and thus the resulting global matrix possesses a block circulant structure. This structure is exploited to develop an efficient matrix decomposition algorithm for the solution of the resulting system. Further savings in the matrix decomposition algorithm are obtained by the use of fast Fourier transforms. The proposed algorithm is used, in conjunction with the method of fundamental solutions for the solution of three-dimensional inhomogeneous elliptic boundary value problems.

Second-order stabilized explicit Runge–Kutta methods for stiff problems

Stabilized Runge–Kutta methods (they have also been called Chebyshev–Runge–Kutta methods) are explicit methods with extended stability domains, usually along the negative real axis. They are easy to use (they do not require algebra routines) and are especially suited for MOL discretizations of two- and three-dimensional parabolic partial differential equations. Previous codes based on stabilized Runge–Kutta algorithms were tested with mildly stiff problems. In this paper we show that they have some difficulties to solve efficiently problems where the eigenvalues are very large in absolute value (over 10 5). We also develop a new procedure to build this kind of algorithms and we derive second-order methods with up to 320 stages and good stability properties. These methods are efficient numerical integrators of very large stiff ordinary differential equations. Numerical experiments support the effectiveness of the new algorithms compared to well-known methods as RKC, ROCK2, DUMKA3 and ROCK4.

Effective computation of complex-shaped quantum-dot structures

The possibility of growing complex-shaped nanodot structures of various material composition allows optimization of certain physical parameters. In the present work, we present effective analytical methods for computing conduction-band eigenstates in quantum-dot structures of complex shape. Comparison with detailed finite-element computations is made. The electronic bandstructure model used is a one-band \(\vec{k}\cdot\vec {p}\) model assuming infinite barriers. Results based on two semi-analytical models are presented. The first model employs geometrical perturbation theory to obtain the quantitative effect of quantum-dot surface perturbations on electron energy levels. Furthermore, the method output includes the level of degeneracy and variations with geometry to be assessed. The second model allows both energy levels and eigenstates to be easily determined for three-dimensional axisymmetrical GaAs structures of varying radius embedded in an AlGaAs matrix by extending a method originally due to Stevenson on electromagnetic waveguide structures (Stevenson in J. Appl. Phys. 22:1447, 1951) to account for electron states. The latter model simplifies the description of a three-dimensional partial-differential equation problem into a small set of ordinary differential equations. For structures with a large aspect ratio, the small set reduces to a single ordinary differential equation yet maintaining high accuracy. A case study is presented to exemplify the models shown.

Electro-diffusion equations of monovalent cations in glass under charge neutrality approximation for optical waveguide fabrication

Two improvements of the usual three-dimensional partial differential equations that model ion-exchanged optical devices in glass under charge neutrality approximation are proposed. First, they are rigorously generalised for both non-ideal cation behaviour and concentration-dependent self-diffusion coefficients. Secondly, the Faraday equation is imposed in the derivation instead of the Ohm law. Accordingly, the current density distribution is not governed by the electric field but by the gradient of an effective electrical potential instead. On the other hand, the boundary conditions of those equations are obtained from standard electrolyte theory, which results in contact potentials at interfaces between glass and molten salt, silver film or metallic cathode. Therefore, typical arrangements for field-assisted ion exchange must be considered as electrochemical cells, and thus a correction term must be added to the applied voltage to get a proper modelling. This term can be neglected in some typical cases but it must be retained in others. Finally, it is predicted that the electrochemical cell effect can be used to diffuse cations from a silver film without need for an external applied field. This suggests a new simple method to fabricate integrated optical components by ion exchange in glass.

Incremental unknowns in three-dimensional stationary problem

The main purpose of this work is to set up the explicit matrix framework appropriate to three-dimensional partial differential equations by means of the incremental unknowns method. Multilevel schemes of the incremental unknowns are presented in the three space dimensions, and through numerical experiments, we confirm that the incremental unknowns method is efficient and the hierarchical preconditioning based on the incremental unknowns can be applied in a more general form.

A Novel Finite-Element– Numerical-Integration Model for Composite Laminates Supported on Opposite Edges

Abstract An attempt is made here to devise a new methodology for an integrated stress analysis of laminated composite plates wherein both in-plane and transverse stresses are evaluated simultaneously. The method is based on the governing three-dimensional (3D) partial differential equations (PDEs) of elasticity. A systematic procedure is developed for a case when one of the two in-plane dimensions of the laminate is considered infinitely long (y direction) with no changes in loading and boundary conditions in that direction. The laminate could then be considered in a two-dimensional (2D) state of plane strain in x-z plane. It is here that the governing 2D PDEs are transformed into a coupled system of first-order ordinary differential equations (ODEs) in transverse z direction by introducing partial discretization in the finite inplane direction x. The mathematical model thus reduces to solution of a boundary value problem (BVP) in the transverse z direction in ODEs. This BVP is then transformed into a set of initial value problems (IVPs) so as to use the available efficient and effective numerical integrators for them. Through thickness displacement and stress fields at the finite element discrete nodes are observed to be in excellent agreement with the elasticity solution. A few new results for cross-ply laminates under clamped support conditions are also presented for future reference and also to show the generality of the formulation.

Hydromagnetic three-dimensional free convective heat transfer overa stretching surface embedded in a non-Darcian porous medium inthe presence of heat generation or absorption

A general analysis is developed to study fluid-flow and heat-transfer characteristics for steady, three-dimensional flow over a linearly stretching porous vertical surface embedded in a non-Darcian medium. A generalized flow model is used in the present study to include the effects of the macroscopic viscous term and the microscopic inertial force. The flow is subjected to a uniform transverse magnetic field normal to the plate. The effect of internal heat generation or absorption is also considered. The governing three-dimensional partial-differential equations for the present case are transformed into ordinary-differential equations using three-dimensional similarity variables. The resulting equations are solved numerically. Velocity distribution, temperature distribution, surface shear stresses, and wall-heat transfer rate are computed for various values of the Prandtl number, magnetic field parameter, inverse Darcy number, porous-medium inertia coefficient, heat generation/absorption coefficient, and mass-transfer coefficient.PACS No.: 44.30.+v

Hydrodymagnetic three-dimensional flow over a stretching surface with heat and mass transfer

A general analysis has been developed to study the combined effect of the free convective heat and mass transfer on the steady three-dimensional laminar boundary layer flow over a stretching surface. The flow is subject to a transverse magnetic field normal to the plate. The governing three-dimensional partial differential equations for the present case are transformed into ordinary differential equation using three-dimensional similarity variables. The resulting equations, are solved numerically by applying a fifth order Runge-Kutta-Fehlberg scheme with the shooting technique. The effects of the Magnetic field Parameter M, buoyancy parameter N, Prandtl number Pr and Schmidt number Sc are examined on the velocity, temperature and concentration distributions. Numerical data for the skin-friction coefficients, Nusselt and Sherwood numbers have been tabulated for various parametric conditions. The results are compared with known from the literature.

A general framework for nonlinear multigrid inversion

A variety of new imaging modalities, such as optical diffusion tomography, require the inversion of a forward problem that is modeled by the solution to a three-dimensional partial differential equation. For these applications, image reconstruction is particularly difficult because the forward problem is both nonlinear and computationally expensive to evaluate. In this paper, we propose a general framework for nonlinear multigrid inversion that is applicable to a wide variety of inverse problems. The multigrid inversion algorithm results from the application of recursive multigrid techniques to the solution of optimization problems arising from inverse problems. The method works by dynamically adjusting the cost functionals at different scales so that they are consistent with, and ultimately reduce, the finest scale cost functional. In this way, the multigrid inversion algorithm efficiently computes the solution to the desired fine-scale inversion problem. Importantly, the new algorithm can greatly reduce computation because both the forward and inverse problems are more coarsely discretized at lower resolutions. An application of our method to Bayesian optical diffusion tomography with a generalized Gaussian Markov random-field image prior model shows the potential for very large computational savings. Numerical data also indicates robust convergence with a range of initialization conditions for this nonconvex optimization problem.

Modelling a spatially heterogeneous biofilm and the bulk fluid: selected results from Benchmark Problem 2 (BM2)

The numerical simulation of mass transfer and conversion in spatially heterogeneous biofilms on the meso-scale requires an accurate description of the hydrodynamics in the biofilm systems and of spatial effects. This leads to systems of three-dimensional nonlinear partial differential equations that are numerically very expensive to solve and to data requirements that are not easy to meet. In this paper several modeling approaches to reduce the physical complexity and, hence, accelerate the computation are compared. They range from a mere reduction of dimensionality by lumping the problem along a secondary flow direction to global mass balances or empirical correlations, at the core of which a one-dimensional boundary value problem must be solved. It is found that even strongly simplified models can describe the qualitative behaviour of the model with regard to variations in the geometrical and hydrodynamic model parameters quite well. In order to obtain also quantitatively reliable results the hydrodynamics must be considered in an appropriate manner.

Hydrodynamic stability of binary mixtures in Bénard and thermogravitational cells

The Soret coefficient of a binary mixture can be measured indirectly by considering its effect on the hydrodynamics induced by free convection. In this paper, we have analyzed the stability of two configurations commonly used for such measurements, the thermogravitational column and the Bénard cell. We have numerically integrated the three-dimensional partial differential equations that describe these systems in order to provide accurate data of stability boundaries suitable for comparison with experimental measurements.