Dirichlet Conditions Research Articles

Revisiting an analytical solution for the three-shell spherical human head model in electroencephalography

The complete three-shell spherical human head model in electroencephalography (EEG) is revisited and an analytical solution of the forward problem is derived. The introduced geometrical model involves four concentric spheres that represent the successive interfaces between the cerebrum, the cerebrospinal fluid, the skull and the skin, which are characterized by different conductivities, while the outer environment is evidently the non-conductive air. The neuronal operation of the brain is considered to be represented by an equivalent and arbitrarily orientated electric dipole that is located in the inner sphere. The dipole source produces a bipolar primary current and the electric activity is initiated by means of the generated electric field, which is associated with the corresponding potential functions within each one of the conductive compartments of the model, inferring crucial information about EEG effects outside the head. The potentials formulae are obtained in compact fashion via the solution of a sequence of interconnected elliptic-type boundary value problems with Dirichlet and Neumann transmission conditions, where the consistent behavior of the fields in the brain and far away from the system has been taken into account. The efficiency of the suggested mathematical model is numerically implemented and the impact of the brain electric response to the exterior measurable potential field is demonstrated, implying a solid and sufficient head representation for the EEG forward problem.

CONTINUOUS-FLOW EXTRACTION OF BIOCOMPOUNDS IN FIXED BED: INFLUENCE OF SWAPPING FROM DIRICHLET TO DANCKWERTS CONDITION AT INLET IN PHENOMENOLOGICAL MODELS

Phenomenological models have increasingly become vital to bioprocess engineering. In continuous-flow biocompounds extraction models, diffusion requires an extra boundary condition at exit (usually null Neumann condition) while either Dirichlet or Danckwerts condition can be imposed at inlet. By taking an extant case study and with the help of an in-house lattice-Boltzmann simulator, this work numerically examines prospective effects of interchanging aforesaid inlet conditions. Trial simulations were performed for scenarios ranging from convective-dominant to diffusive-dominant. Extraction yields numerically simulated under each inlet condition were compared with experimental data. Expected shape of extraction yield curves was simulated whenever process parameters were properly provided and differences due to switching inlet conditions became evident only in diffusion-dominant extraction scenarios. At diffusivities of order 10-6 m2 s-1, numerical results suggest that Danckwerts boundary condition should be preferred at bed inlet.

The Meyers Estimates for Domains Perforated along the Boundary

In this paper, we consider an elliptic problem in a domain perforated along the boundary. By setting a homogeneous Dirichlet condition on the boundary of the cavities and a homogeneous Neumann condition on the outer boundary of the domain, we prove higher integrability of the gradient of the solution to the problem.

Global Existence of Solution for the Fisher Equation via Faedo–Galerkin’s Method

In this study, we consider the Fisher equation in bounded domains. By Faedo–Galerkin’s method and with a homogeneous Dirichlet conditions, the existence of a global solution is proved.

Spectral Galerkin method for solving Helmholtz boundary integral equations on smooth screens

Abstract We solve first-kind Fredholm boundary integral equations arising from Helmholtz and Laplace problems on bounded, smooth screens in three dimensions with either Dirichlet or Neumann conditions. The proposed Galerkin–Bubnov methods take as discretization elements pushed-forward weighted azimuthal projections of standard spherical harmonics onto the unit disk. By exactly depicting edge singular behaviors we show that these spectral or high-order bases yield super-algebraic error convergence in the corresponding energy norms whenever the screen is an analytic deformation of the unit disk. Moreover, we provide a fully discrete analysis of the method, including quadrature rules, based on analytic extensions of the spectral basis to complex neighborhoods. Finally, we include numerical experiments to support our claims as well as appendices with computational details for treating the associated singular integrals.

Parameter dependent finite element analysis for ferronematics solutions

This paper focuses on the analysis of a free energy functional, that models a dilute suspension of magnetic nanoparticles in a two-dimensional nematic well. The first part of the article is devoted to the asymptotic analysis of global energy minimizers in the limit of vanishing elastic constant, ℓ→0 where the re-scaled elastic constant ℓ is inversely proportional to the domain area. The first results concern the strong H1-convergence and a ℓ-independent H2-bound for the global minimizers on smooth bounded 2D domains, with smooth boundary and topologically trivial Dirichlet conditions. The second part focuses on the discrete approximation of regular solutions of the corresponding non-linear system of partial differential equations with cubic non-linearity and non-homogeneous Dirichlet boundary conditions. We establish (i) the existence and local uniqueness of the discrete solutions using fixed point argument, (ii) a best approximation result in energy norm, (iii) error estimates in the energy and L2 norms with ℓ-discretization parameter dependency for the conforming finite element method. Finally, the theoretical results are complemented by numerical experiments on the discrete solution profiles, the numerical convergence rates that corroborates the theoretical estimates, followed by plots that illustrate the dependence of the discretization parameter on ℓ.

Development of a Non-Iterative Macromodeling Technique by Data Integration and Least Square Method

In this paper, a new method is introduced to synthesize the original data obtained from simulation or measurement results in the form of a rational function. The integration of the available data is vital to the performance of the proposed method. The values of poles and residues of the rational model are determined by solving the system of linear equations using the conventional Least Square Method (LSM). To ensure the stability condition of the provided model, a controller coefficient is considered. Also, using this parameter, the designer can increase the stability margin of a system with poor stability conditions. The introduced method has the potential to be used for a wide range of practical applications since there is no specific restriction on the use of this method. The only requirement that should be considered is the Dirichlet condition for the original data, usually the case for physical systems. To verify the performances of the proposed method, several application test cases are investigated and the obtained results are compared with those gathered by the well-known vector fitting algorithm. Also, the examinations show that the method is efficient in the presence of noisy data.

Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part II: recursive computations by the boundary integral equation method

PurposeThe purpose of this paper is to revisit the recursive computation of closed-form expressions for the topological derivative of shape functionals in the context of time-harmonic acoustic waves scattering by sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions).Design/methodology/approachThe elliptic boundary value problems in the singularly perturbed domains are equivalently reduced to couples of boundary integral equations with unknown densities given by boundary traces. In the case of circular or spherical holes, the spectral Fourier and Mie series expansions of the potential operators are used to derive the first-order term in the asymptotic expansion of the boundary traces for the solution to the two- and three-dimensional perturbed problems.FindingsAs the shape gradients of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.Originality/valueThe authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function in the iterated numerical solution of any shape optimization or imaging problem relying on time-harmonic acoustic waves propagation. When coupled with converging Gauss−Newton iterations for the search of optimal boundary parametrizations, it generates fully automatic algorithms.

Green’s functions for fractional difference equations with Dirichlet boundary conditions

This article is devoted to deduce the expression and the main properties of the Green’s function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In particular, it is proved that such function has constant sign on their set of definition and, moreover, it satisfies some additional strong sign conditions that are fundamental to define suitable cones, where to ensure the existence of solutions of nonlinear problems.

On the asymptotic formulas for perturbations in the eigenvalues of the Stokes equations due to the presence of small deformable inclusions

Abstract In this paper, we provide a rigorous derivation of an asymptotic formula for the perturbation of eigenvalues associated to the Stokes eigenvalue problem with Dirichlet conditions and in the presence of small deformable inclusions. Taking advantage of the small sizes of the inclusions immersed in an incompressible Newtonian fluid having kinematic viscosity different from the background one, we show that our asymptotic formula can be expressed in terms of the eigenvalue in the absence of the inclusions and in terms of the viscous moment tensor (VMT).

Patch‐smoother and multigrid for the dual formulation for linear elasticity

AbstractThe dual formulation for linear elasticity, in contrast to the primal formulation, is not affected by locking, as it is based on the stresses as main unknowns. Thus it is quite attractive for nearly incompressible and incompressible materials. Discretization with mixed finite elements will lead to—possibly large—linear saddle point systems. Whereas efficient multigrid methods exist for solving problems in mixed plane elasticity for nearly incompressible materials, we propose a multigrid method that is also stable in the incompressible limit. There are two main challenges in constructing a multigrid method for the dual formulation for linear elasticity. First, in the incompressible limit, the matrix block related to the stress is positive semidefinite. Second, the stress belongs to and standard smoothers, working for regular problems, cannot be applied. We present a novel patch‐based smoother for the dual formulation for linear elasticity. We discuss different types of local boundary conditions for the patch subproblems. Based on our patch‐smoother, we build a multigrid method for the solution of the resulting saddle point problem and investigate its efficiency and robustness. Numerical experiments show that Dirichlet and Robin conditions work best and eventually lead to textbook multigrid performance.

The solution of arbitrary smoothness of the one-dimensional wave equation for the problem with mixed conditions

In this paper, we represented an analytical form of a classical solution of the wave equation in the class of continuously differentiable functions of arbitrary order with mixed boundary conditions in a quarter of the plane. The boundary of the area consists of two perpendicular half-lines. On one of them, the Cauchy conditions are specified. The second half-line is separated into two parts, namely, the limited segment and the remaining part in the form of a half-line. The Dirichlet condition is specified on the segment, as well as the Neumann condition is fulfilled on the second part in the form of a half-line. In a quarter of the plane, the classical solution of the problem under consideration is determined. To construct this solution, a particular solution of the original wave equation is established. For the given functions of the problem, the concordance conditions are written, which are necessary and sufficient for the solution of the problem to be classical of high order of smoothness and unique.

The Friedland–Hayman inequality and Caffarelli’s contraction theorem

The Friedland–Hayman inequality is a sharp inequality concerning the growth rates of homogeneous, harmonic functions with Dirichlet boundary conditions on complementary cones dividing Euclidean space into two parts. In this paper, we prove an analogous inequality in which one divides a convex cone into two parts, placing Neumann conditions on the boundary of the convex cone and Dirichlet conditions on the interface. This analogous inequality was already proved by us jointly with Sarah Raynor. Here, we present a new proof that permits us to characterize the case of equality. In keeping with the two-phase free boundary theory introduced by Alt, Caffarelli, and Friedman, such an improvement can be expected to yield further regularity in free boundary problems.

Diffraction of EM-wave by a slit of finite width with Dirichlet conditions in cold plasma

In the present communication, we have carefully analyzed the interaction of incident wave field on a slit of finite width under the imposition of Dirichlet boundary conditions in the presence of cold plasma. The boundary value problem is established after implementing the Fourier transform on Helmholtz equation. The Wiener-Hopf analysis is utilized to sort out the problem proposed for this model. The separated field is evaluated and its numerical results are presented, at the end of the analysis, to further study the effects of various physical parameters in the presence of cold plasma.

An L2-Poincaré–Dolbeault lemma of spaces with mixed cone-cusp singular metrics

The existence of Kähler Einstein metrics with mixed cone and cusp singularity has received considerable attentions in recent years. It is believed that such kind of metric would give rise to important geometric invariants. We computed their [Formula: see text]-Hodge–Frölicher spectral sequence under the Dirichlet and Neumann boundary conditions and examine the pure Hodge structures on them. It turns out that these cohomologies agree well with the de Rham cohomology of a good compactification.

Homogenization and Correctors for Stochastic Hyperbolic Equations in Domains with Periodically Distributed Holes

The goal of this paper is to present new results on homogenization and correctors for stochastic linear hyperbolic equations in periodically perforated domains with homogeneous Neumann conditions on the holes. The main tools are the periodic unfolding method, energy estimates, probabilistic and deterministic compactness results. The findings of this paper are stochastic counterparts of the celebrated work [D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Port. Math. (N.S.) 63 (2006) 467–496]. The convergence of the solution of the original problem to a homogenized problem with Dirichlet condition has been shown in suitable topologies. Homogenization and convergence of the associated energies results recover the work in [M. Mohammed and M. Sango, Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains, Asymptot. Anal. 97 (2016) 301–327]. In addition to that, we obtain corrector results.

A Nitsche stabilized finite element method: Application for heat and mass transfer and fluid–structure interaction

A Nitsche stabilized finite element method is proposed for heat & mass transfer and fluid–structure interaction. The GLS/PSPG stabilization is employed to stabilize the finite element formulation. The Nitsche’s methods are employed to weakly impose the Dirichlet condition for heat & mass transfer. An upwind term is included in Nitsche’s methods to enhance the stability for fast moving interfaces. The Ghost penalty method is employed to control the jumps across the cut cells. The projection-based adaptive Gauss quadrature (PAGQ) scheme is used for the numerical integration of the discontinuous function. The nonlinear advection–diffusion equations are linearized by Newton procedure. The second-order accurate unconditionally stable generalized-α time integration is implemented to march the solution in time. The fluid and the structure equations are weakly coupled by a second-order accurate staggered-partitioned scheme. Numerical examples include the cases of fixed/vibrating/rotating cylinder(s) for heat & mass transfer and fluid–structure interaction in enclosure and external flow. The obtained numerical results match well with the experiments, the empirical correlation and the numerical simulations in literature. This methodology efficiently and accurately simulates the complex physics of heat & mass transfer.

Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part I: the free space case

PurposeIn this paper, the authors revisit the computation of closed-form expressions of the topological indicator function for a one step imaging algorithm of two- and three-dimensional sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions) in the free space.Design/methodology/approachFrom the addition theorem for translated harmonics, explicit expressions of the scattered waves by infinitesimal circular (and spherical) holes subject to an incident plane wave or a compactly supported distribution of point sources are available. Then the authors derive the first-order term in the asymptotic expansion of the Dirichlet and Neumann traces and their surface derivatives on the boundary of the singular medium perturbation.FindingsAs the shape gradient of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.Originality/valueThe authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function that generates initial guesses in the iterated numerical solution of any shape optimization problem or imaging problems relying on time-harmonic acoustic wave propagation.

The Hybrid FEM-DBCI for the Solution of Open-Boundary Low-Frequency Problems

This paper describes a particular use of the hybrid FEM-DBCI, for the computation of low-frequency electromagnetic fields in open-boundary domains. Once the unbounded free space enclosing the system has been truncated, the FEM is applied to the bounded domain thus obtained, assuming an unknown Dirichlet condition on the truncation boundary. An integral equation is used to express this boundary condition in which the integration surface is selected in the middle of the most external layer of finite elements, very close to the truncation boundary, so that the integral equation becomes quasi-singular. The method is described for the computation of electrostatic fields in 3D and of eddy currents in 2D, but it is also applicable to the solution of other kinds of electromagnetic problems. Comparisons are made with other methods, concluding that FEM-DBCI is competitive with the well-known FEM-BEM and coordinate transformations for what concerns accuracy and computing time.

Solving Poisson equation with Dirichlet conditions through multinode Shepard operators

The multinode Shepard operator is a linear combination of local polynomial interpolants with inverse distance weighting basis functions. This operator can be rewritten as a blend of function values with cardinal basis functions, which are a combination of the inverse distance weighting basis functions with multivariate Lagrange fundamental polynomials. The key for simply computing the latter, on a unisolvent set of points, is to use a translation of the canonical polynomial basis and the PA=LU factorization of the associated Vandermonde matrix. In this paper, we propose a method to numerically solve a Poisson equation with Dirichlet conditions through multinode Shepard interpolants by collocation. This collocation method gives rise to a collocation matrix with many zero entrances and a smaller condition number with respect to the one of the well known Kansa method. Numerical experiments show the accuracy and the performance of the proposed collocation method.