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Laplace Equation (laplace + equation)
Selected AbstractsIdentification and separation of DNA-hybridized nanocolloids by Taylor cone harmonicsELECTROPHORESIS, Issue 18 2009Xinguang Cheng Abstract A rapid (minutes) electrospray bead-based DNA hybridization detection technique is developed by spraying a mixture of hybridized and unhybridized silica nanocolloids. With proper far-field control by external electrodes, the trajectory of the ejected nanobeads from the electrospray is governed by specific harmonics of the Laplace equation, which select discrete polar angles along well-separated field maxima near the conducting Taylor cone. Due to Rayleigh fission and evaporation, beads of different size acquire different total charge after ejection and suffer different normal electrophoretic displacement such that they are ejected along well-separated field maxima and are deposited in distinct rings on an intersecting plane. As the hybridized DNA is of the same dimension as that of the nanocolloid, the nanocolloids are hence easily differentiated from the unhybridized ones. This technique is highly specific as the high shear stress in the microjet shears away any non-specifically bound DNA from the nanocolloid surface. [source] Volume of a liquid drop detaching from a sphereHEAT TRANSFER - ASIAN RESEARCH (FORMERLY HEAT TRANSFER-JAPANESE RESEARCH), Issue 6 2010Kenji Katoh Abstract A theoretical and experimental study is conducted to investigate the detached volume from a pendant drop on the surface of a sphere. Observation of drop detachment by high-speed video camera reveals that the movement of the upper part of the neck of the drop is quite slow compared to that of the detaching lower part. The surface profile of the upper part was calculated approximately as a static problem using the axisymmetric Laplace equation. Using the drop profile, the system energy, including the work done by the solid,liquid wetting behavior, was calculated. Based on the condition of minimum energy, the volume of the detached part V was calculated. The volume V increases with the sphere diameter and approaches the value for the pendant drop attached to a plate. In addition, V is strongly dependent on the wettability between the sphere and the liquid and decreases with the receding contact angle. The detached volume of the water drop was measured for spheres of porous brick of various diameters. The experimental and theoretical results were found to be in good agreement. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/htj.20305 [source] The boundary element method for solving the Laplace equation in two-dimensions with oblique derivative boundary conditionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2007D. Lesnic Abstract In this communication, we extend the Neumann boundary conditions by adding a component containing the tangential derivative, hence producing oblique derivative boundary conditions. A variant of Green's formula is employed to translate the tangential derivative to the fundamental solution in the boundary element method (BEM). The two-dimensional steady-state heat conduction with the imposed oblique boundary condition has been tested in smooth, piecewise smooth and multiply connected domains in which the Laplace equation is the governing equation, producing results at the boundary in excellent agreement with the available analytical solutions. Convergence of the normal and tangential derivatives at the boundary is also achieved. The numerical boundary data are then used to successfully calculate the values of the solution at interior points again. The outlined test cases have been repeated with various boundary element meshes, indicating that the accuracy of the numerical results increases with increasing boundary discretization. Copyright © 2006 John Wiley & Sons, Ltd. [source] Least-square-based radial basis collocation method for solving inverse problems of Laplace equation from noisy dataINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2010Xian-Zhong Mao Abstract The inverse problem of 2D Laplace equation involves an estimation of unknown boundary values or the locations of boundary shape from noisy observations on over-specified boundary or internal data points. The application of radial basis collocation method (RBCM), one of meshless and non-iterative numerical schemes, directly induces this inverse boundary value problem (IBVP) to a single-step solution of a system of linear algebraic equations in which the coefficients matrix is inherently ill-conditioned. In order to solve the unstable problem observed in the conventional RBCM, an effective procedure that builds an over-determined linear system and combines with least-square technique is proposed to restore the stability of the solution in this paper. The present work investigates three examples of IBVPs using over-specified boundary conditions or internal data with simulated noise and obtains stable and accurate results. It underlies that least-square-based radial basis collocation method (LS-RBCM) poses a significant advantage of good stability against large noise levels compared with the conventional RBCM. Copyright © 2010 John Wiley & Sons, Ltd. [source] Local discretization error bounds using interval boundary element methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2009B. F. Zalewski Abstract In this paper, a method to account for the point-wise discretization error in the solution for boundary element method is developed. Interval methods are used to enclose the boundary integral equation and a sharp parametric solver for the interval linear system of equations is presented. The developed method does not assume any special properties besides the Laplace equation being a linear elliptic partial differential equation whose Green's function for an isotropic media is known. Numerical results are presented showing the guarantee of the bounds on the solution as well as the convergence of the discretization error. Copyright © 2008 John Wiley & Sons, Ltd. [source] Explicit expressions for 3D boundary integrals in potential theory,INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2009S. Nintcheu Fata Abstract On employing isoparametric, piecewise linear shape functions over a flat triangular domain, exact expressions are derived for all surface potentials involved in the numerical solution of three-dimensional singular and hyper-singular boundary integral equations of potential theory. These formulae, which are valid for an arbitrary source point in space, are represented as analytic expressions over the edges of the integration triangle. They can be used to solve integral equations defined on polygonal boundaries via the collocation method or may be utilized as analytic expressions for the inner integrals in the Galerkin technique. In addition, the constant element approximation can be directly obtained with no extra effort. Sample problems solved by the collocation boundary element method for the Laplace equation are included to validate the proposed formulae. Published in 2008 by John Wiley & Sons, Ltd. [source] Discrete duality finite volume schemes for two-dimensional drift-diffusion and energy-transport modelsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2009C. Chainais-Hillairet Abstract The drift-diffusion and the energy-transport models appear in the modelling of semiconductor devices. The main difficulty arising in the approximation of the energy transport model by finite volume schemes is the discretization of the Joule heating term in the equation on the density of energy. Following some recent ideas by Domelevo and Omnès for the discretization of the Laplace equation on almost general meshes, we construct a finite volume approximation of the 2-D drift-diffusion and energy transport models. These schemes still hold on almost general meshes. Finally, we present numerical simulations of semiconductor devices. Copyright © 2006 John Wiley & Sons, Ltd. [source] Stress Development Due to Capillary Condensation in Powder Compacts: A Two-Dimensional Model StudyJOURNAL OF THE AMERICAN CERAMIC SOCIETY, Issue 6 2000Stefan Lampenscherf A model experiment is presented to investigate the relationship between the humidity-dependent liquid distribution and the macroscopic stress in a partially wet powder compact. Therefore, films of monosized spherical particles were cast on silicon substrates. Using environmental SEM the geometry of the liquid necks trapped between particles was imaged as a function of relative humidity. Simultaneously the macroscopic stress in the substrate adhered particle film was measured by capacitive deflection measurement. The experimentally found humidity dependence of the liquid neck size and the macroscopic film stress are compared with model predictions. The circle,circle approximation is used to predict the size of the liquid necks between touching particles as a function of the capillary pressure. Using the modified Kelvin relation between capillary pressure and relative humidity, we consider the effect of an additional solute which may be present in the capillary liquid. The results of the stress measurement are compared with the model predictions for a film of touching particles in hexagonal symmetry. The contribution of the capillary interaction to the adhesion force between neighboring particles is calculated using the integrated Laplace equation. The resulting film stress can be approximated relating this capillary force to an effective cross section per particle. The experimentally found humidity dependence of the liquid neck size is in good agreement with the model predictions for finite solute concentration. The film stress corresponds to the model predictions only for large relative humidities and shows an unexpected increase at small values. As is shown with an atomic force microscope, the real structure of the particle,particle contact area changes during the wet/dry cycle. A solution/reprecipitation process causes surface heterogeneities and solid bridging between the particles. It is claimed that the existence of a finite contact zone between the particles gives rise to the unexpected increase of the stress at small relative humidities. [source] Heat transfer at high energy devices with prescribed cooling flowMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2007Jens Breuer Abstract We study the heat transfer from a high-energy electric device into a surrounding cooling flow. We analyse several simplifications of the model to allow an easier numerical treatment. First, the flow variables velocity and pressure are assumed to be independent from the temperature which allows a reduction to Prandtl's boundary layer model and leads to a coupled nonlinear transmission problem for the temperature distribution. Second, a further simplification using a Kirchhoff transform leads to a coupled Laplace equation with nonlinear boundary conditions. We analyse existence and uniqueness of both the continuous and discrete systems. Finally, we provide some numerical results for a simple two-dimensional model problem. Copyright © 2006 John Wiley & Sons, Ltd. [source] The ice-fishing problem: the fundamental sloshing frequency versus geometry of holesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2004Vladimir Kozlov Abstract We study an eignevalue problem with a spectral parameter in a boundary condition. This problem for the Laplace equation is relevant to sloshing frequencies that describe free oscillations of an inviscid, incompressible, heavy fluid in a half-space covered by a rigid dock with some apertures (an ice sheet with fishing holes). The dependence of the fundamental eigenvalue on holes' geometry is investigated. We give conditions on a plane region guaranteeing that the fundamental eigenvalue corresponding to this region is larger than the fundamental eigenvalue corresponding to a single circular hole. Examples of regions satisfying these conditions and having the same area as the unit disk are given. New results are also obtained for the problem with a single circular hole. On the other hand, we construct regions for which the fundamental eigenfrequency is larger than the similar frequency for the circular hole of the same area and even as large as one wishes. In the latter examples, the hole regions are either not connected or bounded by a rather complicated curves. Copyright © 2004 John Wiley & Sons, Ltd. [source] On finite difference potentials and their applications in a discrete function theoryMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16-18 2002K. Gürlebeck Abstract We present a potential theoretical method which is based on the approximation of the boundary value problem by a finite difference problem on a uniform lattice. At first the discrete fundamental solution of the Laplace equation is studied and the theory of difference potentials is described. In the second part we define a discrete Cauchy integral operator and a Teodorescu transform. In addition a Borel,Pompeiu formula can be formulated. Copyright © 2002 John Wiley & Sons, Ltd. [source] Approximate identities in variable Lp spacesMATHEMATISCHE NACHRICHTEN, Issue 3 2007D. Cruz-Uribe SFO Abstract We give conditions for the convergence of approximate identities, both pointwise and in norm, in variable Lp spaces. We unify and extend results due to Diening [8], Samko [18] and Sharapudinov [19]. As applications, we give criteria for smooth functions to be dense in the variable Sobolev spaces, and we give solutions of the Laplace equation and the heat equation with boundary values in the variable Lp spaces. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] For the inf-sup condition on mesh-dependent norms on a nonconvex polygonNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2008Sungkyu Choi Abstract It is shown that the inf-sup condition, called the Babuska,Brezzi condition, is valid for certain mesh-dependent norms on a nonconvex polygonal domain. A bilinear form that is derived by inserting the corner singularity expansion into the Laplace equation is considered. A mesh-dependent fractional norm related to the least order of the corner singularity at a concave vertex is considered. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source] Image analysis using p -Laplacian and geometrical PDEsPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2007A. KuijperArticle first published online: 29 FEB 200 Minimizing the integral ,,1/p |,L |pd , for an image L under suitable boundary conditions gives PDEs that are well-known for p = 1, 2, namely Total Variation evolution and Laplacian diffusion (also known as Gaussian scale space and heat equation), respectively. Without fixing p, one obtains a framework related to the p -Laplace equation. The partial differential equation describing the evolution can be simplified using gauge coordinates (directional derivatives), yielding an expression in the two second order gauge derivatives and the norm of the gradient. Ignoring the latter, one obtains a series of PDEs that form a weighted average of the second order derivatives, with Mean Curvature Motion as a specific case. Both methods have the Gaussian scale space in common. Using singularity theory, one can use properties of the heat equation (namely. the role of scale) in the full L (x, t) space and obtain a framework for topological image segmentation. In order to be able to extend image analysis aspects of Gaussian scale space in future work, relations between these methods are investigated, and general numerical schemes are developed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Numerical simulation of non-viscous liquid pinch off using a coupled level set boundary integral methodPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2007Maria Garzon The pinch off of an inviscid fluid column is described using a potential flow model with capillary forces. The interface velocity is obtained via a Galerkin boundary integral method for the 3D axisymmetric Laplace equation, whereas the interface location and the velocity potential on the free boundary are both approximated using level set techniques on a fixed domain. The algorithm is validated computing the Raleigh-Taylor instability for liquid columns which provides an analytical solution for short times. The simulations show the time evolution of the fluid tube and the algorithm is capable of handling pinch-off and after pinch-off events. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Ohmic heating of dairy fluids,effects of local electric field on temperature distributionASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING, Issue 5 2009Heng Jin Tham Abstract This paper presents the heat transfer model of a continuous flow ohmic heating process. The model fluid used was a mixture of reconstituted skimmed milk and whey protein concentrate solution. Two-dimensional numerical simulations of an annular ohmic heater were performed using a general purpose partial differential equation solver, FlexPDE. The momentum, energy, and electrical equations were solved for a laminar flow regime. Two models were used to determine the volumetric heating rate, one taking into account the local electric field by solving the Laplace equation while another model assumes an average voltage gradient applied between the two electrodes. Results show that while the wall temperature distribution is different for the two cases, the bulk fluid temperature and the average outlet temperature are the same. The predicted temperatures generally agree well with the measured temperatures. Copyright © 2009 Curtin University of Technology and John Wiley & Sons, Ltd. [source] Power concavity on nonlinear parabolic flowsCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2005Ki-Ahm Lee Our object in this paper is to show that the concavity of the power of a solution is preserved in the parabolic p -Laplace equation, called power concavity, and that the power is determined by the homogeneity of the parabolic operator. In the parabolic p -Laplace equation for the density u, the concavity of u(p,2)/p is considered, which indicates why the log-concavity has been considered in heat flow, p = 2. In addition, the long time existence of the classical solution of the parabolic p -Laplacian equation can be obtained if the initial smooth data has -concavity and a nondegenerate gradient along the initial boundary. © 2004 Wiley Periodicals, Inc. [source] FETI-DP, BDDC, and block Cholesky methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2006Jing Li Abstract The FETI-DP and BDDC algorithms are reformulated using Block Cholesky factorizations, an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations. Instead of introducing Lagrange multipliers to enforce the coarse level, primal continuity constraints in these algorithms, a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom. With the new formulation of these algorithms, a simplified proof is provided that the spectra of a pair of FETI-DP and BDDC algorithms, with the same set of primal constraints, are essentially the same. Numerical experiments for a two-dimensional Laplace's equation also confirm this result. Copyright © 2005 John Wiley & Sons, Ltd. [source] A hybrid boundary node methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2002Jianming Zhang Abstract A new variational formulation for boundary node method (BNM) using a hybrid displacement functional is presented here. The formulation is expressed in terms of domain and boundary variables, and the domain variables are interpolated by classical fundamental solution; while the boundary variables are interpolated by moving least squares (MLS). The main idea is to retain the dimensionality advantages of the BNM, and get a truly meshless method, which does not require a ,boundary element mesh', either for the purpose of interpolation of the solution variables, or for the integration of the ,energy'. All integrals can be easily evaluated over regular shaped domains (in general, semi-sphere in the 3-D problem) and their boundaries. Numerical examples presented in this paper for the solution of Laplace's equation in 2-D show that high rates of convergence with mesh refinement are achievable, and the computational results for unknown variables are most accurate. No further integrations are required to compute the unknown variables inside the domain as in the conventional BEM and BNM. Copyright © 2001 John Wiley & Sons, Ltd. [source] Self-regular boundary integral equation formulations for Laplace's equation in 2-DINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2001A. B. Jorge Abstract The purpose of this work is to demonstrate the application of the self-regular formulation strategy using Green's identity (potential-BIE) and its gradient form (flux-BIE) for Laplace's equation. Self-regular formulations lead to highly effective BEM algorithms that utilize standard conforming boundary elements and low-order Gaussian integrations. Both formulations are discussed and implemented for two-dimensional potential problems, and numerical results are presented. Potential results show that the use of quartic interpolations is required for the flux-BIE to show comparable accuracy to the potential-BIE using quadratic interpolations. On the other hand, flux error results in the potential-BIE implementation can be dominated by the numerical integration of the logarithmic kernel of the remaining weakly singular integral. Accuracy of these flux results does not improve beyond a certain level when using standard quadrature together with a special transformation, but when an alternative logarithmic quadrature scheme is used these errors are shown to reduce abruptly, and the flux results converge monotonically to the exact answer. In the flux-BIE implementation, where all integrals are regularized, flux results accuracy improves systematically, even with some oscillations, when refining the mesh or increasing the order of the interpolating function. The flux-BIE approach presents a great numerical sensitivity to the mesh generation scheme and refinement. Accurate results for the potential and the flux were obtained for coarse-graded meshes in which the rate of change of the tangential derivative of the potential was better approximated. This numerical sensitivity and the need for graded meshes were not found in the elasticity problem for which self-regular formulations have also been developed using a similar approach. Logarithmic quadrature to evaluate the weakly singular integral is implemented in the self-regular potential-BIE, showing that the magnitude of the error is dependent only on the standard Gauss integration of the regularized integral, but not on this logarithmic quadrature of the weakly singular integral. The self-regular potential-BIE is compared with the standard (CPV) formulation, showing the equivalence between these formulations. The self-regular BIE formulations and computational algorithms are established as robust alternatives to singular BIE formulations for potential problems. Copyright © 2001 John Wiley & Sons, Ltd. [source] A Lagrangian boundary element approach to transient three-dimensional free surface flow in thin cavitiesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2001Jie Zhang Abstract The lubrication theory is extended for transient free-surface flow of a viscous fluid inside a three-dimensional thin cavity. The problem is closely related to the filling stage during the injection molding process. The pressure, which in this case is governed by the Laplace's equation, is determined using the boundary element method. A fully Lagrangian approach is implemented for the tracking of the evolving free surface. The domain of computation is the projection of the physical domain onto the (x,,y) plane. This approach is valid for simple and complex cavities as illustrated for the cases of a flat plate and a curved plate. It is found that the flow behavior is strongly influenced by the shape of the initial fluid domain, the shape of the cavity, and inlet flow pressure. Copyright © 2001 John Wiley & Sons, Ltd. [source] Altered T Wave Dynamics in a Contracting Cardiac ModelJOURNAL OF CARDIOVASCULAR ELECTROPHYSIOLOGY, Issue 2003NICOLAS P. SMITH Ph.D. Introduction: The implications of mechanical deformation on calculated body surface potentials are investigated using a coupled biophysically based model. Methods and Results: A cellular model of cardiac excitation-contraction is embedded in an anatomically accurate two-dimensional transverse cross-section of the cardiac ventricles and human torso. Waves of activation and contraction are induced by the application of physiologically realistic boundary conditions and solving the bidomain and finite deformation equations. Body surface potentials are calculated from these activation profiles by solving Laplace's equation in the passive surrounding tissues. The effect of cardiac deformation on electrical activity, induced by contraction, is demonstrated in both single-cell and tissue models. Action potential duration is reduced by 7 msec when the single cell model is subjected to a 10% contraction ramp applied over 400 msec. In the coupled electromechanical tissue model, the T wave of the ECG is shown to occur 18 msec earlier compared to an uncoupled excitation model. To assess the relative effects of myocardial deformation on the ECG, the activation sequence and tissue deformation are separated. The coupled and uncoupled activation sequences are mapped onto the undeforming and deforming meshes, respectively. ECGs are calculated for both mappings. Conclusion: Adding mechanical contraction to a mathematical model of the heart has been shown to shift the T wave on the ECG to the left. Although deformation of the myocardium resulting from contraction reduces the T wave amplitude, cell stretch producing altered cell membrane kinetics is the major component of this temporal shift. (J Cardiovasc Electrophysiol, Vol. 14, pp. S203-S209, October 2003, Suppl.) [source] | ||||||||||||||||