Differential Equations In Domains Research Articles

Output characteristics modeling and experimental verification of secondary-uncompensated inductive power delivery link operating without feedback

Series-none compensated induction wireless power delivery link (SN-IPDL) was recently shown to be capable of achieving minimal complexity receiver with reduced losses in case of strong coupling, typical for applications such as power delivery into enclosed compartments such as industrial glove boxes and close-range inductive heating. Common practical applications often feature power delivery to varying loads which require a stable operational voltage. Operation at LIVO frequency allows establishing potentially feedback-free wireless power delivery with constant output voltage for varying loads. The paper reveals that due to nonzero equivalent series resistances and invalidity of the first-harmonic-approximation (FHA) based equivalent circuit, the output voltage remains affected by the load despite operation at LIVO frequency, residing within a certain range of values, lower and upper bounds of which correspond to rated and zero loading, respectively. This must be considered during SN-IPDL system design in order to assure that output voltage stays within allowed bounds for relevant loading conditions. This paper derives analytical expressions for output voltage bounds of a power-loaded series-to SN-IPDL, functioning at load-independent-voltage-output (LIVO) frequency. Time domain differential equations (DE)-based analysis is applied to derive the two bounds and allows (as a by-product) establishing an alternative FHA-based equivalent circuit of SN-IPDL operating in continuous conduction mode. Simulations and experiments based on 380V, 1.2 kW-rated prototype demonstrate excellent matching with analytical outcomes, validating the proposed analysis.

Method of Fictitious Domains and Homotopy as a New Alternative to Multidimensional Partial Differential Equations in Domains of Any Shape

The ideas of the method of fictitious domains and homotopy are combined with an aim to reduce the solution of boundary-value problems for multidimensional partial differential equations (PDE) in domains of any shape to an exponentially convergent sequence of PDE in a parallelepiped (or, in the 2D case, in a rectangle). This enables us to decrease the required computer time due to the elimination of the necessity of triangulation of the domain by a grid with N inner nodes (thus, the Delaunay algorithm in the 2D case requires $$ \mathcal{O} $$ (N log N) operations).

Equation—Domain Duality in the Dirichlet Problem for General Differential Equations in the Space L2

A development of an author’s observation that led to the creation of the equation—domain duality method is presented. This method is used in the study of the Dirichlet problem for a general partial differential equation in a semialgebraic domain. The exposition involves results of the general theory of boundary value problems and is aimed at extending these results to the generalized statements of such problems in L2(Ω). Results on the boundary properties of the L2-solution of a general linear partial differential equation in a domain are employed. It is demonstrated how the general construction under consideration is used in the study of the Dirichlet problem for specific equations with constant coefficients on the basis of the equation—domain duality method. It is also shown how one can extend to the generalized statement of the Dirichlet problem the earlier obtained necessary and sufficient conditions for the existence of a nontrivial smooth solution to the homogeneous Dirichlet problem for a general second-order equation with constant complex coefficients and a homogeneous symbol in a disk, as well as for an ultrahyperbolic equation in the n-dimensional ball.

The fictitious domain method and homotopy as a new alternative for multidimensional partial differential equations in domains of arbitrary shape

The fictitious domain method and homotopy as a new alternative for multidimensional partial differential equations in domains of arbitrary shape

Smoothed Boundary Method for Diffusion-Related Partial Differential Equations in Complex Geometries

The smoothed boundary method has emerged as an attractive method for solving partial differential equations in domains of complex morphology. In the current work, we present two schemes for incorporating Dirichlet boundary conditions on complex domains into partial differential equations through an energy penalty formulation of the smoothed boundary method, which may be straightforwardly combined with, e.g., variational phase-field methods. Convergence to the exact solution is guaranteed in a certain limit, and it is shown how parameters of the smoothed boundary method should be chosen to minimize error, and avoid additional error introduced by poor spatial resolution during numerical solution. An example demonstrating the simplicity of the implementation of the methods is presented, and the scheme believed to have most general merit is determined.

Hierarchical model reduction for incompressible fluids in pipes

SummaryHierarchical model reduction is intended to solve efficiently partial differential equations in domains with a geometrically dominant direction. In many engineering applications, these problems are often reduced to 1‐dimensional differential systems. This guarantees computational efficiency yet dumps local accuracy as nonaxial dynamics are dropped. Hierarchical model reduction recovers the secondary components of the dynamics of interest with a combination of different discretization techniques, following up a natural separation of variables. The dominant direction is generally solved by the finite element method or isogeometric analysis to guarantee flexibility, while the transverse components are solved by spectral methods, to guarantee a small number of degrees of freedom. By judiciously selecting the number of transverse modes, the method has been proven to improve significantly the accuracy of purely 1‐dimensional solvers, with great computational efficiency. A Cartesian framework has been used so far both in slab domains and cylindrical pipes (including arteries) mapped to Cartesian reference domains. In this paper, we investigate the alternative use of a polar coordinates system for the transverse dynamics in circular or elliptical pipes. This seems a natural choice for applications like computational hemodynamics. In spite of this, the selection of a basis function set for the transverse dynamics is troublesome. As pointed out in the literature—even for simple elliptical problems—there is no “best” basis available. In this paper, we perform an extensive investigation of hierarchical model reduction in polar coordinates to discuss different possible choices for the transverse basis, pointing out pros and cons of the polar coordinate system.

Análise da convergência da solucao de squacoes lineares elípticas sob um esquema de diferencas finitas generalizadas (MDFG)

This work prove the convergence of Generalized Finite Difference Method (MDFG), presented as a alternative meshless method to the solution of nonlinear elliptic differential equations. This analysis presented in this work is a theoretical contribution of (MDFG) as an alternative method for the solution of differential equations in domains with irregular geometric. The analysis result shows an linear truncation error of the method for two-dimensional linear problems using a polynomial second-degree approach.

On domain symmetry and its use in homogenization

The present paper focuses on solving partial differential equations in domains exhibiting symmetries and periodic boundary conditions for the purpose of homogenization. We show in a systematic manner how the symmetry can be exploited to significantly reduce the complexity of the problem and the computational burden. This is especially relevant in inverse problems, when one needs to solve the partial differential equation (the primal problem) many times in an optimization algorithm. The main motivation of our study is inverse homogenization used to design architected composite materials with novel properties which are being fabricated at ever increasing rates thanks to recent advances in additive manufacturing. For example, one may optimize the morphology of a two-phase composite unit cell to achieve isotropic homogenized properties with maximal bulk modulus and minimal Poisson ratio. Typically, the isotropy is enforced by applying constraints to the optimization problem. However, in two dimensions, one can alternatively optimize the morphology of an equilateral triangle and then rotate and reflect the triangle to form a space filling D3 symmetric hexagonal unit cell that necessarily exhibits isotropic homogenized properties. One can further use this D3 symmetry to reduce the computational expense by performing the “unit strain” periodic boundary condition simulations on the single triangle symmetry sector rather than the six fold larger hexagon. In this paper we use group representation theory to derive the necessary periodic boundary conditions on the symmetry sectors of unit cells. The developments are done in a general setting, and specialized to the two-dimensional dihedral symmetries of the abelianD2, i.e. orthotropic, square unit cell and nonabelian D3, i.e. trigonal, hexagon unit cell. We then demonstrate how this theory can be applied by evaluating the homogenized properties of a two-phase planar composite over the triangle symmetry sector of a D3 symmetric hexagonal unit cell.

Elliptic functional differential equations with incommensurable contractions

The “commensurability” of transformations has been a crucial assumption in the study of solvability and regularity of solutions for elliptic functional differential equations in domains, while equations with incommensurable transformations are much less studied. In the paper, we consider an equation containing multiplicatively incommensurable contractions of the arguments of the unknown function in the principal part. Algebraic conditions for unique solvability of the Dirichlet problem will be obtained as well as conditions ensuring the existence of an infinite-dimensional null-space. The equation considered is an elliptic analog of the generalized pantograph equation studied by many authors. As a complementary conclusion, we observe that the spectral properties of functional operators with contractions are unstable with respect to small perturbations of scaling parameters.

Hermitian finite element complementing the Bogner–Fox–Schmit rectangle near curvilinear boundary

We discuss the Hermitian finite elements of high-order accuracy for solving boundary value problems for partial differential equations in domains with curvilinear boundaries. New elements are constructed in such a way that they can be used in conjunction with the Bogner–Fox–Schmit rectangular elements.

A reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks

In this paper we propose a reduced basis hybrid method (RBHM) for the approximation of partial differential equations in domains represented by complex networks where topological features are recurrent. The RBHM is applied to Stokes equations in domains which are decomposable into smaller similar blocks that are properly coupled.The RBHM is built upon the reduced basis element method (RBEM) and it takes advantage from both the reduced basis methods (RB) and the domain decomposition method. We move from the consideration that the blocks composing the computational domain are topologically similar to a few reference shapes. On the latter, representative solutions, corresponding to the same governing partial differential equations, are computed for different values of some parameters of interest, representing, for example, the deformation of the blocks. A generalized transfinite mapping is used in order to produce a global map from the reference shapes of each block to any deformed configuration.The desired solution on the given original computational domain is recovered as projection of the previously precomputed solutions and then glued across subdomain interfaces by suitable coupling conditions.The geometrical parametrization of the domain, by transfinite mapping, induces non-affine parameter dependence: an empirical interpolation technique is used to recover an approximate affine parameter dependence and a subsequent offline/online decomposition of the reduced basis procedure. This computational decomposition yields a considerable reduction of the problem complexity. Results computed on some combinations of 2D and 3D geometries representing cardiovascular networks show the advantage of the method in terms of reduced computational costs and the quality of the coupling to guarantee continuity of both stresses, pressure and velocity at subdomain interfaces.

Fast algorithms and MATLAB software for solution of the Dirichlet boundary value problems for elliptic partial differential equations in domains with complicated geometry

New fast algorithms for solution of the Dirichlet boundary value problem for the class of elliptic Partial Differential Equations (PDE) is proposed. Algorithms are based on new version of General R...

A generalized Ritz method for partial differential equations in domains of arbitrary geometry using global shape functions

A boundary element method (BEM)-based variational method is presented for the solution of elliptic PDEs describing the mechanical response of general inhomogeneous anisotropic bodies of arbitrary geometry. The equations, which in general have variable coefficients, may be linear or nonlinear. Using the concept of the analog equation of Katsikadelis the original equation is converted into a linear membrane (Poisson) or a linear plate (biharmonic) equation, depending on the order of the PDE under a fictitious load, which is approximated with radial basis function series of multiquadric (MQ) type. The integral representation of the solution of the substitute equation yields shape functions, which are global and satisfy both essential and natural boundary conditions, hence the name generalized Ritz method. The minimization of the functional that produces the PDE as the associated Euler–Lagrange equation yields not only the Ritz coefficients but also permits the evaluation of optimal values for the shape parameters of the MQs as well as optimal position of their centers, minimizing thus the error. If a functional does not exists or cannot be constructed as it is the usual case of nonlinear PDEs, the Galerkin principle can be applied. Since the arising domain integrals are converted into boundary line integrals, the method is boundary-only and, therefore, it maintains all the advantages of the pure BEM. Example problems are studied, which illustrate the method and demonstrate its efficiency and great accuracy.

ℋ-matrix approximation for elliptic solution operators in cylinder domains

We develop a data-sparse and accurate approximation of the normalised hyperbolic operator sine family generated by a strongly P-positive elliptic operator defined in [Gavrilyuk, J. Math. Anal. Appl. 236: 327–349, 1999, Gavrilyuk and Makarov, Numer. Funct. Anal. Optimization 20: 695–717, 1999]. In the preceding papers [Hackbusch, Computing 62: 89–108, 1999–Hackbusch, Khoromskij, and Sauter, On ℋ 2 -matrices, Springer-Verlag, 2000], a class of ℋ-matrices has been analysed which is data-sparse and allow an approximate matrix arithmetic with almost linear complexity. An ℋ-matrix approximation to the operator exponent with a strongly P-positive operator was proposed in [Gavrilyuk, Hackbusch, and Khoromskij, ℋ-matrix approximation for the operator exponential with applications, 2000]. In the present paper, we apply the ℋ-matrix techniques to approximate the elliptic solution operator on cylinder-like domains Ω × [ a, b ] associated with the elliptic operator , x ∈ [ a, b ]. It is explicitly presented by the normalised hyperbolic sine of an elliptic operator ℒ defined in Ω. The approach is then applied to elliptic partial differential equations in domains composed of rectangles or cylinders. In particular, we consider the ℋ-matrix approximation to the interface Poincaré-Steklov operators with application in the Schur-complement domain decomposition method. Starting with the Dunford–Cauchy representation for the hyperbolic sine operator, we then discretise the integral by the exponentially convergent quadrature rule involving a short sum of resolvents. The latter are approximated by the ℋ-matrix techniques. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [ a, b ].

New parallel stochastic global integral and local differential equation modeling and inversion

In this paper, a new parallel modeling and inversion algorithm using a Stochastic Global Integral and Local Differential equation (SGILD) decomposition is presented. We derived new acoustic integral equations and differential equations for statistical moments of the parameters and field. The new statistical moment integral equations on the boundary and local differential equations in domain will be used together to obtain mean wave field and its moments in the modeling. The new moment global Jacobian volume integral equations and the local Jacobian differential equations in domain will be used together to update the mean parameters and their moments in the inversion. A new parallel multiple hierarchy substructure direct algorithm or direct-iteration hybrid algorithm will be used to solve the sparse matrices and one smaller full matrix from domain to the boundary, in parallel. The SGILD modeling and imaging algorithm has many advantages over the conventional imaging approaches. The SGILD algorithm can be used for the stochastic acoustic, electromagnetic, and flow modeling and inversion.

A computer science approach for solving elliptic differential equations

AbstractA method that combines computer science and numerical analysis techniques for the solution of elliptic partial differential equations in domains consisting of the union of rectangles is presented. It is applied to Laplace's equation and to the Navier–Stokes equations in several domains. The method is very flexible and can be easily modified to solve other equations. © 1995 John Wiley & Sons, Inc.

Limit boundary value problems in regions with random fine grained boundaries

We consider stochastic partial differential equations in domains with a random boundary. For the solution, we use earlier results with respect to solutions of nonstochastic partial differential equations with random boundaries, from one side, and solutions of stochastic partial differential equations with nonrandom boundaries, on the other side. The last problem is related to the extrapolation of stochastic fields.

Averaging of elliptic systems of differential equations in domains with holes. II. The case of a periodic distribution of holes

Averaging of elliptic systems of differential equations in domains with holes. II. The case of a periodic distribution of holes

Series expansions of solutions of plane elliptic partial differential equations in domains of arbitrary shape

In this work we use function - theoretic methods to construct geometry-dependent bilinear series expansions of the fundamental solutions of plane elliptic partial differential equations with entire coefficients, convergent throughout interior or exterior domains of aritrary shape. Using these bilinear expansions we construct a Faber-type series expansion of the general solution of the equation, convergent throughout any given bounded multiply-connected domain. Each term in this expansion is given explicitly via a contour integral representation involving the fundamental solution and a confor-mal mapping function associated with a simply-connected domain

Asymptotic expansions for the coefficient functions that arise in turning-point problems

We study the uniform asymptotic expansion for a large parameter u of solutions of second-order linear differential equations in domains of the complex plane which include a turning point. The standard theory of Olver demonstrates the existence of three such expansions, corresponding to solutions of the form Ai ( u 2/3 ζ e 2/3απi ) Ʃ n s = 0 A s (ζ)/ u 2 s + u -2 d / d ζ Ai ( u 2/3 ζ e 2/3απi ) Ʃ n -1 s =0 B s (ζ)/u 2 s + ϵ ( α ) n ( u , ζ) for α = 0, 1, 2, with bounds on ϵ ( α ) n . We proceed differently, by showing that the set of all solutions of the differential equation is of the form Ai ( u 2/3 ζ) A ( u , ζ) + u -2 (d/dζ) Ai ( u 2/3 ζ) B ( u , ζ), where Ai denotes any situation of Airy's equation. The coefficent functions A ( u , ζ) and B ( u , ζ) are the focus of our attention : we show that for sufficiently large u they are holomorphic functions of ζ in a domain including the turning point, and as functions of u are described asymptotically by descending power series in u 2 , with explicit error bounds. We apply our theory to Bessel functions.