Integral Equation Formulation Research Articles

On the Domain Decomposition Method Preconditioning of Surface Integral Equation Formulations Solved by GMRES

On the Domain Decomposition Method Preconditioning of Surface Integral Equation Formulations Solved by GMRES

APPLICATION OF THE POTENTIAL THEORY TO SOLVING A MIXEDPROBLEM FOR A MULTIDIMENSIONAL ELLIPTIC EQUATION WITHONE SINGULAR COEFFICIENT

Potential theory has played a paramount role in both analysis and computation for boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in integral equation form. By using some properties of Gaussian hypergeometric function, we prove limiting theorems and derive integral equations concerning a densities of the double- and simple-layer potentials. The obtained results are applied to find a solution of the mixed problem for the multidimensional singular elliptic equation in the half-space

DETERMINATION OF THE DROP SIZE AND DISTRIBUTIONS IN SWIRL INJECTION IN CROSS FLOWS, IMPINGING, AND EFFERVESCENT INJECTORS

We have extended the primary atomization analysis to swirl injection in cross flows, impinging, and effervescent injectors. Using the integral form of the conservation equations, the drop size can be expressed in terms of injection and fluid parameters, the main variable being the liquid and gas velocities. Using the measured velocities as inputs to this <i>D</i><sub>32</sub>-equation, good agreements with experimental data are found for the drop size in the three spray geometries. Underlying physical mechanisms for the drop formation are also revealed from the analysis. The aerodynamic interaction between the swirl spray and cross flow results in reduction in momentum, with a corresponding decrease in kinetic energy that appears as surface tension of energy of many small droplets. Similarly, cancellation of the lateral momentum in impinging jets and internal deceleration in effervescent injectors are the key primary atomization routes. The use of the analytical drop size-velocity correlation has also been demonstrated for swirl sprays in cross flows. Therefore, this approach can be used to predict the drop size and distributions in different spray geometries, with appropriate changes in the velocity input terms and fluid properties.

Stochastic Modeling in Immunology Based On a Stage-Dependent Framework with Non-Markov Constraints for Individual Cell and Pathogen Dynamics

We present a systematic approach to modelling the responses of the immune system to virus infections. Two continuous-discrete stochastic models arising in mathematical immunology are developed and computationally implemented. The variables of the models are integer random variables that denote the quantity of individuals (cells and viral particles), and sets of unique types of individuals that take into account the current state and history of stay of individuals in some stages of their development. The distribution laws of the durations of the mentioned stages are different from exponential or geometric. A probabilistic description of a one-stage stochastic model of population dynamics is presented. A stochastic model of the development of HIV-1 infection in the lymph node in the initial period after infection of a healthy person is formulated. A computational algorithm based on the Monte Carlo method is given. Each of the stochastic models is complemented by a deterministic analogue in the form of integral and delay differential equations. The results of numerical simulation are presented.

Mathematical micro–macro modeling of fully coupled nonlinear magneto-elastic reinforced composites

In this paper, integral equation formulations and field-dependent micromechanical modeling of global nonlinear magneto-mechanical behavior of magneto-elastic composites under high strain and magnetic field are elaborated. The modeling is obtained based on an extension of micro–macro transition inclusion problem to nonlinear behavior using field-dependent and highly nonlinear localization tensors. A methodological procedure is elaborated based on newly introduced strain and magnetic field-dependent Green tensors, strain and magnetic field-dependent integral equations linked to tensors of Eshelby and micromechanical approaches. The field-dependent global moduli are predicted based on the Mori–Tanaka approach and self-consistent method. Iterative incremental schemes based on the Newton–Raphson and fixed-point algorithms are examined and established precise semi-analytic equations of the effective magneto-elastic properties of composites that are dependent on the magnetic-strain field for different types of inclusions. A numerical code is elaborated for numerical predictions and the obtained field-dependent effective magneto-elastic coefficients are obtained and presented for various volume fractions of inclusion, types and shapes of the reinforced nonlinear composites.

A Generalized Source Integral Equation for Enhanced Compression in Three Dimensions

An integral equation formulation, using generalized directional sources, for three-dimensional scattering by impenetrable and essentially convex bodies, is presented. This formulation increases the rank-deficiency of moment matrix blocks representing interactions between non-overlapping source and observer regions. Thereby, it enables enhanced low-rank approximation based matrix compression and the development of corresponding fast direct solvers. The directional sources are constructed by augmenting the conventional basis functions with spherical absorbing three-dimensional shields, on which auxiliary source distributions are defined. The bottleneck arising from the need to integrate over these distributions upon computing the modified Green’s function is removed by using efficient non-uniform sampling and tabulation of the modified Green’s function or its components, in a region-dependent manner. The formulation is studied and its favorable compressibility is demonstrated, for two fundamental types of compression strategies. The non-uniform sampling approach is employed also to facilitate the rank-revealing analysis of large off-diagonal blocks.

Efficient Calculation of the 3-D Rectangular Waveguide Green’s Functions Derivatives by the Ewald Method

In this contribution, the Ewald method has efficiently been applied to accelerate the computation of the rectangular waveguide Green’s functions (GFs) derivatives. Based on previous works, we have outlined new approximation formulae that avoid the evaluation of computationally expensive complementary error functions of complex argument, needed by the Ewald method. This is possible when the internal medium of the rectangular waveguide is homogeneous and lossless. On the other hand, different convergence numerical studies have been carried out, showing a similar convergence rate for computing the original GFs and their derivatives. Moreover, we have checked that the computational time is only slightly increased for obtaining the derivatives as compared to the original GFs, after the application of these new techniques. The newly derived expressions are useful for the evaluation of electromagnetic fields, the characterization of dielectric materials and step discontinuities between rectangular waveguides, and the analysis of rectangular cavities using integral equation formulations. For validation, the electric field produced by a surface electric current density with a rectangular pulse distribution has been evaluated, using the new proposed expressions. These results have been compared to simulations provided by full-wave finite elements commercial software to verify their correctness, exhibiting a good agreement.

Existence and stability of solution for a nonlinear Volterra integral equation with binary relation via fixed point results

The real-world problems that can be resolved by a dynamic mathematical model is an analogous form of a complex integral equation. In this article, we intend to exhibit the solution to a non-homogeneous, nonlinear Volterra integral equation endowed with an arbitrary binary relation utilizing the fixed point solution that is originally proved for a multivalued map and then reduced to a single valued self map. The results proved and the application demonstrated is an added asset to the present literature and can significantly contribute in determining the existence of a solution for a dynamic mathematical model in the form of a Volterra integral equation equipped with an arbitrary binary relation via fixed point findings. Furthermore, the stability of the solution of the Volterra integral equation is verified using the Hyers–Ulam stability concept.

Size-dependent nonlinear forced vibration of microbeam using two-phase local/nonlocal viscoelastic integral model with thermal effects

Combination strain-driven (εD) and stress-driven (σD) two-phase local/nonlocal integral models (TPNIM) with classic Kelvin-Voigt viscoelastic model, εD- and σD-two-phase local/nonlocal viscoelastic integral models (TPNVIM) are proposed to study the viscoelastic nonlinear forced vibration behaviors of microbeams in thermal environment. Based on von-Karman large deformation theory and the Hamilton’s principle, the differential governing equations of motion and boundary conditions are derived for nonlinear forced vibration of microbeam. Several non-dimensional variables are introduced to simplify the mathematical formulation. Nonlocal viscoelastic equations in integral form are transferred unitedly into equivalent differential form with constitutive constraints. The axial force of nonlocal microbeam with immovable ends is derived explicitly. Vibration frequency and corresponding linear vibration mode shape are derived through Laplace transformation for linear thermo-elastic vibration of microbeam using εD- and σD-TPNIMs. The Ritz-Galerkin approach is utilized to derive the reduced differential governing equation for nonlinear vibration of microbeam, which has exact same form as single-degree-of-freedom oscillator with cubic nonlinearity when material viscosity is neglected. The method of multiple scales (MMS) based on trigonometric functions is utilized to solve the reduced nonlinear differential equation approximately. The frequency-amplitude responses are addressed for primary and super-harmonic resonances of microbeam. The effects of viscous coefficient, amplitude of excitation force and environmental temperature variation as well as nonlocal length parameter on the nonlinear vibration behavior of microbeam are investigated numerically for different boundary conditions.

Nyström discretizations of boundary integral equations for the solution of 2D elastic scattering problems

We present three high-order Nyström discretization strategies of various boundary integral equation formulations of the impenetrable time-harmonic Navier equations in two dimensions. One class of such formulations is based on the four classical Boundary Integral Operators (BIOs) associated with the Green’s function of the Navier operator. We consider two types of Nyström discretizations of these operators, one that relies on Kussmaul–Martensen logarithmic splittings (Chapko et al., 2000; Domínguez and Turc, 2000), and the other on Alpert quadratures (Alpert, 1999). In addition, we consider an alternative formulation of Navier scattering problems based on Helmholtz decompositions of the elastic fields (Dong et al., 2021), which can be solved via a system of boundary integral equations that feature integral operators associated with the Helmholtz equation. Owing to the fact that some of the BIOs that are featured in those formulations are non-standard, we use Quadrature by Expansion (QBX) methods for their high order Nyström discretization. Alternatively, we use Maue integration by parts techniques to recast those non-standard operators in terms of single and double layer Helmholtz BIOs whose Nyström discretizations is amenable to the Kussmaul–Martensen methodology. We present a variety of numerical results concerning the high order accuracy that our Nyström discretization elastic scattering solvers achieve for both smooth and Lipschitz boundaries. We also present extensive comparisons regarding the iterative behavior of solvers based on different integral equations in the high frequency regime. Finally, we illustrate how some of the Nyström discretizations we considered can be incorporated seamlessly into the Convolution Quadrature (CQ) methodology to deliver high-order solutions of the time domain elastic scattering problems.

Spectral boundary integral equation method for simulation of 2D and 3D slip ruptures at bi‐material interfaces

AbstractThis paper presents a 3D spectral numerical scheme that can accurately model planarcracks of any shape residing on an interfacial plane between two elastic solids. The cracks propagate dynamically under the action of interfacial loads and interfacial constitutive laws. The scheme is a spectral form of the boundary integral equations (BIE) that relate the displacement discontinuity fields along the interfacial plane to the stress fields at the plane. In the BIE, the displacement discontinuities are expressed as a spatio‐temporal convolution of the traction components at the interface. This is in contrast with most previous studies which use a spatio‐temporal convolution of the displacement discontinuities or of the displacements at the interface. Due to the continuity of tractions across the interface, the present formulation is simpler, resulting in convolution kernels that can be written in closed‐form. Previous studies have relied on numerically obtained convolution kernels. The accuracy of the proposed scheme is validated with the known analytical solution of the 3D Lamb problem. Furthermore, new algorithms are developed for coupling the BIE with a slip‐weakening friction law, both for homogeneous materials as well as for bi‐material interfaces. This results in a widely applicable formulation for studying spontaneous rupture propagation. The algorithms are validated by comparing rupture simulation exercises to benchmark problems from Southern California Earthquake Center (SCEC). Finally, the methodology is used to simulate 3D frictional rupture propagation along a bi‐material interface.

Fractional approach for Dirac operator involving M-truncated derivative

In this study, we examine the basic spectral information for systems governed by the Dirac equation with distinct boundary conditions, utilizing a modified form of local derivatives known as M-truncated derivative (MTD). The spectral information discussed includes the representation of solutions in the form of integral equations, the asymptotics vector-valued eigenfunctions and eigenvalues, and their normalized forms, all within the context of the MTD method that incorporates truncated Mittag-Leffler functions. This type of MTD provides the features of integer-order operator theory. Also, by virtue of the parameters $\alpha $ and $\gamma$, we analyze and compare the solutions with graphs in terms of different potentials, different eigenvalues and different orders. Thus, the aim of this article is to consider spectral structure of Dirac system in frame of M-truncated derivative by proping with visual analysis.

An SIE Formulation With Triangular Discretization and Loop Analysis for Parameter Extraction of Arbitrarily Shaped Interconnects

A surface integral equation (SIE) formulation under the magneto-quasi-static assumption is proposed to efficiently and accurately model arbitrarily shaped interconnects in packages. Through decently transferring all electromagnetic quantities into circuit elements, the loop analysis is used to carefully construct matrix equations with an independent and complete set of unknowns based on graph theory. In addition, an efficient preconditioner is developed, and the proposed formulation is accelerated by the pre-corrected fast Fourier transform (pFFT). Four practical examples, including a rectangular metallic interconnect, bonding wire arrays, interconnects in a real-life circuit, and the power distribution network (PDN) used in packages, are carried out to validate its accuracy, efficiency, and scalability. Results show that the proposed formulation is accurate, efficient, and flexible to model complex interconnects in packages.

The measurement of true initial rates is not always absolutely necessary to estimate enzyme kinetic parameters

In the chapters dealing with enzyme reactions, the authors of all Biochemistry textbooks and of even more specialized texts consider that the characteristic parameters (kcat and Km) must be determined under initial or steady-state rate conditions. This implies the transformation of a very limited proportion of substrate (at most 10–20%) or a continuous recording of the product or substrate concentration vs. time. Both options can present practical difficulties. Is it possible to get around these very stringent conditions? Here we show that in the most favourable cases up to 70% of the substrate can be converted resulting in systematic errors on the parameters (that can easily be taken account of) if the simple Henri-Michaelis–Menten equation is utilised. Alternatively, the integrated form of the same equation directly yields excellent estimates of the same parameters. Our observations should greatly facilitate the task of researchers who study systems in which measurements of the reaction progress are painstaking or when substrate concentrations close to the detection limit must be used. The general conclusion is that it is not always absolutely necessary to determine initial or steady-state rates to obtain reliable estimations of the enzyme kinetic parameters..

Critical local well‐posedness for the fully nonlinear Peskin problem

AbstractWe study the problem where a one‐dimensional elastic string is immersed in a two‐dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non‐linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time wellposedness for arbitrary initial data in the scaling critical Besov space . We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancelation structure.

Multiscaling Differential Contraction Integral Method for Inverse Scattering Problems With Inhomogeneous Media

Practical applications of microwave imaging often require the solution of inverse scattering problems with inhomogeneous backgrounds. Toward this end, a novel inversion strategy, which combines the multiscaling (MS) regularization scheme and the difference contraction integral equation (DCIE) formulation, is proposed. Such an integrated approach mitigates the nonlinearity and the ill-posedness of the problem to obtain reliable high-resolution reconstructions of the unknown scattering profiles. The arising algorithmic implementation, denoted as MS-DCIE, does not require the computation of Green’s function of the inhomogeneous background, and thus, it provides an efficient and effective way to deal with complex scenarios. The performance of the MS-DCIE is assessed by means of numerical and experimental tests in comparison with competitive state-of-the-art inversion strategies as well.

A Matrix Transform Technique for Distributed-Order Time-Fractional Advection–Dispersion Problems

Invoking the matrix transfer technique, we propose a novel numerical scheme to solve the time-fractional advection–dispersion equation (ADE) with distributed-order Riesz-space fractional derivatives (FDs). The method adopts the midpoint rule to reformulate the distributed-order Riesz-space FDs by means of a second-order linear combination of Riesz-space FDs. Then, a central difference approximation is used side by side with the matrix transform technique for approximating the Riesz-space FDs. Based on this, the distributed-order time-fractional ADE is transformed into a time-fractional ordinary differential equation in the Caputo sense, which has an equivalent Volterra integral form. The Simpson method is used to discretize the weakly singular kernel of the resulting Volterra integral equation. Stability, convergence, and error analysis are presented. Finally, simulations are performed to substantiate the theoretical findings.

Data-driven analysis of parametrized acoustic systems in the frequency domain

A data-driven method combined with the formulations of boundary integral equations is developed for the frequency-domain analysis of parametrized acoustic systems, arising from the spatial discretization of the linear Helmholtz equation. The method derives surrogate models for the approximation of frequency response functions at selected field points via the construction of neural networks with radial basis function interpolation. This requires the offline collection of a training database consisting of discrete frequency-parameter samples and their associated response values, whereas the information about system matrices and interference with boundary element kernels are no longer necessary, leading to a matrix-free and non-intrusive nature. To ensure accurate fitting results, a tensor decomposition technique is employed to decouple the naturally formed tensor grid between the frequency and parameter inputs in the database. After establishing two sets of radial basis networks separately, the final reduced-order solutions as a continuous function of the frequency and parameters are represented by the linear combinations of tensor products. In the online phase, the original transfer function at any testing frequency-parameter location can be quickly predicted as direct output from the compact surrogate model. With the analytical solution, the traditional boundary element method and the fast multipole boundary element method as solvers for the generation of training data, two benchmark problems and one complex system are investigated. In all cases, the simplicity, versatility and efficiency of the proposed method are demonstrated through examples of multi-frequency solutions of acoustic systems with parametrized material and geometric properties.

Finite Element Discretizations for Variable-Order Fractional Diffusion Problems

We present a finite element discretization scheme for multidimensional fractional diffusion problems with spatially varying diffusivity and fractional order. We consider the symmetric integral form of these nonlocal equations defined on general geometries and in arbitrary bounded domains. A number of challenges are encountered when discretizing these equations. The first comes from the heterogeneous kernel singularity in the fractional integral operator. The second comes from the formally dense discrete operator with its quadratic growth in memory footprint and arithmetic operations. An additional challenge comes from the need to handle volume conditions—the generalization of classical local boundary conditions to the nonlocal setting. Satisfying these conditions requires that the effect of the whole domain, including both the interior and exterior regions, be computed on every interior point in the discretization. Performed directly, this would result in quadratic complexity. In order to address these challenges, we propose a strategy that decomposes the stiffness matrix into three components. The first is a sparse matrix that handles the singular near-field separately, and is computed by adapting singular quadrature techniques available for the homogeneous case to the case of spatially variable order. The second component handles the remaining smooth part of the near-field as well as the far-field, and is approximated by a hierarchical {mathcal {H}}^2 matrix that maintains linear complexity in storage and operations. The third component handles the effect of the global mesh at every node, and is written as a weighted mass matrix whose density is computed by a fast-multipole type method. The resulting algorithm has therefore overall linear space and time complexity. Analysis of the consistency of the stiffness matrix is provided and numerical experiments are conducted to illustrate the convergence and performance of the proposed algorithm.

Integral equation methods for the Morse-Ingard equations

We present two (a decoupled and a coupled) integral-equation-based methods for the Morse-Ingard equations subject to Neumann boundary conditions on the exterior domain. Both methods are based on second-kind integral equation (SKIE) formulations. The coupled method is well-conditioned and can achieve high accuracy. The decoupled method has lower computational cost and more flexibility in dealing with the boundary layer; however, it is prone to the ill-conditioning of the decoupling transform and cannot achieve as high accuracy as the coupled method. We show numerical examples using a Nyström method based on quadrature-by-expansion (QBX) with fast-multipole acceleration. We demonstrate the accuracy and efficiency of the solvers in both two and three dimensions with complex geometry.