Rectangular Domain Research Articles

Boundary-to-solution mapping for groundwater flows in a Toth basin

In this paper, the authors propose a new approach to solving the groundwater flow equation in the Toth basin of arbitrary top and bottom topographies using deep learning. Instead of using traditional numerical solvers, they use a DeepONet to produce the boundary-to-solution mapping. This mapping takes the geometry of the physical domain along with the boundary conditions as inputs to output the steady state solution of the groundwater flow equation. To implement the DeepONet, the authors approximate the top and bottom boundaries using truncated Fourier series or piecewise linear representations. They present two different implementations of the DeepONet: one where the Toth basin is embedded in a rectangular computational domain, and another where the Toth basin with arbitrary top and bottom boundaries is mapped into a rectangular computational domain via a nonlinear transformation. They implement the DeepONet with respect to the Dirichlet and Robin boundary condition at the top and the Neumann boundary condition at the impervious bottom boundary, respectively. Using this deep-learning enabled tool, the authors investigate the impact of surface topography on the flow pattern by both the top surface and the bottom impervious boundary with arbitrary geometries. They discover that the average slope of the top surface promotes long-distance transport, while the local curvature controls localized circulations. Additionally, they find that the slope of the bottom impervious boundary can seriously impact the long-distance transport of groundwater flows. Overall, this paper presents a new and innovative approach to solving the groundwater flow equation using deep learning, which allows for the investigation of the impact of surface topography on groundwater flow patterns.

Homogenization of elliptic PDE with L 1 source term in domains with boundary having very general oscillations

In the present article, we study the homogenization of a second-order elliptic PDE with oscillating coefficients in two different domains, namely a standard rectangular domain with very general oscillations and a circular type oscillating domain. Further, we consider the source term in L 1 and hence the solutions are interpreted as renormalized solutions. In the first domain, oscillations are in horizontal directions, while that of the second one is in the angular direction. To take into account the type of oscillations, we have used two different types of unfolding operators and have studied the asymptotic behavior of the renormalized solution of a second-order linear elliptic PDE with a source term in L 1 . In fact, we begin our study in oscillatory circular domain with oscillating coefficients and L 2 data which is also new in the literature. We also prove relevant strong convergence (corrector) results. We present the complete details in the context of circular domains, and sketch the proof in other domain.

An interface preserving and residual-based adaptivity for phase-field modeling of fully Eulerian fluid-structure interaction

In this paper, we present a novel interface preserving and residual-based adaptivity procedure for the phase-field modeling of a fully Eulerian fluid-structure interaction. The proposed adaptive fully Eulerian procedure involves a fixed background unstructured finite element mesh on which the fluid-structure boundary is treated implicitly via a diffused interface description. We employ the stabilized finite element formulation and a partitioned iterative procedure to solve the coupled system of the Allen-Cahn phase-field equation with the unified momentum equation comprising solid and fluid dynamics. To evaluate the solid stresses, the left Cauchy-Green deformation tensor is convected at each time step to trace the evolution of the solid strain in the Eulerian reference frame. We utilize a residual based error indicator and the newest vertex bisection algorithm for the adaptive refinement/coarsening of the unstructured mesh. Through the adaptive finite element method, we aim to preserve the profile of the diffused interface in the presence of convective distortion. We propose an early stopping criterion for the adaptivity of the interface dynamics by combining the effects of the phase-field interface thickness parameter and the mesh resolution. In addition, we control the size of the refined/coarsened elements by introducing a characteristic length scale governed by a weighted harmonic mean of the desired bulk and the interface mesh resolutions. The proposed stopping criterion and the characteristic length scale provide significant reductions in the computational cost while simultaneously ensuring convergence properties of the coupled fluid-structure system. We perform a detailed convergence and accuracy analysis via two benchmark problems namely, the pure solid system and a coupled fluid-solid system with an interface in a rectangular domain. We further provide a comparative study between feature- and residual based adaptive strategies. We systematically assess the performance of the adaptive procedure in terms of conservation properties and computational efficiency. Finally, we demonstrate our fully Eulerian adaptive solver to simulate the contact and bouncing phenomenon between an elastic solid and a rigid wall.

Example of a solution for Dorodnitzyn’s limit formula

In the present paper, we show an example of a solution for Dorodnitzyn’s gaseous boundary layer limit formula. Oleinik’s no back-flow condition ensures the existence and uniqueness of solutions for the Prandtl equations in a rectangular domain R⊂R2. It also allowed us to find a limit formula for Dorodnitzyn’s stationary compressible boundary layer with constant total energy on a bounded convex domain in the plane R2. Under the same assumption, we can give an approximate solution u for the limit formula if |u|<<<1 such that: u(z)≅δ∗c∗z+625⋅12i0⋅4U23z4+o(z5),that corresponds to an approximate horizontal velocity component when a small parameter ϵ given by the quotient of the maximum height of the domain divided by its length tends to zero. Here, c>0, δ is the boundary layer’s height in Dorodnitzyn’s coordinates, U is the free-stream velocity at the upper boundary of the domain, and T0 is the absolute surface temperature.

Boundary-Value Problem for the Fourth-Order Equation with Multiple Characteristics in a Rectangular Domain

For a fourth-order equation with minor terms, we consider a boundary-value problem in a rectangular domain. The uniqueness of solution of the posed problem is proved by the method of energy integrals. The solution is expressed in terms of the constructed Green function. In the substantiation of uniform convergence, we establish the fact that the “small denominator” is not equal to zero.

FFT‐based efficient Poisson solver in nonrectangular domain

AbstractPoisson's equation is one of the most popular partial differential equation (PDE), which is widely used in image processing, computer graphics and other fields. However, solving a large‐scale Poisson's equation always costs huge computational resources. Fast Fourier transform (FFT) is an efficient Poisson solver but it only works in rectangular domain. In this paper, we propose a FFT‐based Poisson solver in nonrectangular domain on regular grids combined with algebraic multigrid (AMG). We extend the original Poisson's equation to a rectangular domain to construct an equivalent equation, so that it can use FFT algorithm to accelerate the solving to Poisson's equation. Experiments show that the FFT‐based Poisson solver can improve the solving speed of large‐scale Poisson's equations in nonrectangular domain. We demonstrate the solver in applications of image processing and fluid simulation.

Multi-objective topology optimisation for acoustic porous materials using gradient-based, gradient-free, and hybrid strategies.

When designing passive sound-attenuation structures, one of the challenging problems that arise is optimally distributing acoustic porous materials within a design region so as to maximise sound absorption while minimising material usage. To identify efficient optimisation strategies for this multi-objective problem, several gradient, non-gradient, and hybrid topology optimisation strategies are compared. For gradient approaches, the solid-isotropic-material-with-penalisation method and a gradient-based constructive heuristic are considered. For gradient-free approaches, hill climbing with a weighted-sum scalarisation and a non-dominated sorting genetic algorithm-II are considered. Optimisation trials are conducted on seven benchmark problems involving rectangular design domains in impedance tubes subject to normal-incidence sound loads. The results indicate that while gradient methods can provide quick convergence with high-quality solutions, often gradient-free strategies are able to find improvements in specific regions of the Pareto front. Two hybrid approaches are proposed, combining a gradient method for initiation and a non-gradient method for local improvements. An effective Pareto-slope-based weighted-sum hill climbing is introduced for local improvement. Results reveal that for a given computational budget, the hybrid methods can consistently outperform the parent gradient or non-gradient method.

On the Dirichlet Problem in Rectangular Domain for Lavrentiev–Bitsadze Equation

On the Dirichlet Problem in Rectangular Domain for Lavrentiev–Bitsadze Equation

An almost second-order robust computational technique for singularly perturbed parabolic turning point problem with an interior layer

The subject of this paper is the numerical investigation of a singularly perturbed parabolic problem with a simple interior turning point on a rectangular domain in the x−t plane with Dirichlet boundary conditions. At the location of the turning point, the solution to this class of problems has an internal layer. By decomposing the solution into a regular and singular component, we may first establish stronger bounds for the regular and singular components, as well as their derivatives. The fitted mesh finite difference scheme, which consists of a hybrid finite difference operator on a non-uniform Shishkin mesh in the space direction and a backward implicit Euler scheme on a uniform mesh in the time variable, is used to identify the approximate numerical solution. The accuracy of the resulting scheme is then improved in the temporal direction using the Richardson extrapolation scheme, which reveals that the extrapolated scheme is almost second-order uniformly convergent in both space and time variables when ɛ≤CN−1. Two numerical examples are used to demonstrate the efficacy and accuracy of the finite difference approaches mentioned.

Solvability of a Nonlocal Inverse Problem for a Fourth-Order Equation

В данной работе для уравнения в частных производных четвертого порядка в прямоугольной области рассмотрена нелокальная обратная задача по поиску неизвестной правой части, которая зависит от одной переменной. Собственные функции и присоединенные функции соответствующей спектральной задачи, и их биортогональные функции полны и образуют базис Рисса в пространстве L2​(0,1). Установлены критерии единственности и существования решения рассматриваемой нелокальной обратной задачи для уравнения четвертого порядка. Единственность решения нелокальной обратной задачи вытекает из полноты системы биортогональных функций. Решение задачи построено в виде суммы ряда по собственным и присоединенным функциям, соответствующей спектральной задачи. Установлены достаточные условия на граничные функции, которые гарантируют теоремы существования и устойчивости решения рассматриваемой задачи. В замкнутой области показана абсолютная и равномерная сходимость найденного решения обратной задачи в виде ряда, а также рядов, полученных почленным дифференцированием по t и x соответственно два и четыре раза, в зависимости от гладкости функции заданными начальными условиями. При этом возникают малые знаменатели, затрудняющие сходимость этих рядов. Доказано, что в зависимости от размера области, множество ненулевых решений выражения в знаменателе не пусто. А также, показано, что если этот знаменатель равен нулю, то данная задача будет иметь нетривиальное решение при однородных условиях. Доказана также, устойчивость решения обратной задачи по нормам пространств L2​,W2n​ и C(Ω±), относительно изменения входных данных. In this paper, we consider a nonlocal inverse problem of finding an unknown right-hand side with one variable for a fourth-order partial differential equation in a rectangular domain. The eigenfunctions and functions of the corresponding spectral problem and its biorthogonal functions are complete and form a Riesz basis in the space L2​(0,1). Criteria for the uniqueness and existence of a solution to the considered nonlocal inverse problem for a fourth-order equation are established. The uniqueness of the solution of the nonlocal inverse problem follows from the completeness of the system of biorthogonal functions. The solution of the problem is constructed as the sum of a series in terms of eigenfunctions and associated functions of the corresponding spectral problem. Sufficient conditions are established for the boundary functions that guarantee existence and stability theorems for the solution of the problem under consideration. In a closed domain, absolute and uniform convergence of the found solution of the inverse problem is shown in the form of a series, as well as series obtained by term-by-term differentiation with respect to t and x two and four times, respectively, depending on the smoothness of the function given the initial conditions. In this case, small denominators arise, which hinder the convergence of these series. It is proved that, depending on the size of the domain, the set of non-zero solutions of the expression in the denominator is not empty. And also, it is shown that if this denominator is equal to zero, then this problem will have a non-trivial solution under homogeneous conditions. It is also proved that the solution of the inverse problem is stable in the norms of the spaces L2​,W2n​ and C(Ω±) with respect to changes in the input data.

Нелокальная начально-граничная задача для вырождающиегося уравнения четвертого порядка с дробной производной Герасимова-Капуто

В последнее время интенсивно изучаются начально – граничные задачи в прямоугольной области для дифференциальных уравнений в частных производных как четного, так и нечетного порядка. При этом в качестве объекта исследования, в основном, берется не вырождающееся уравнение или уравнение, вырождающееся на одной стороне четырехугольника. Начально – граничные задачи (как локальные, так и нелокальные) для уравнений с двумя или тремя линиями вырождения остаются неизученными. В данной работе в прямоугольной области рассмотрено уравнение четвёртого порядка, вырождающееся на трех сторонах четырехугольника и содержащее оператор дробного дифференцирования Герасимова –Капуто. Для этого уравнения сформулирована и исследована одна начально – граничная задача с нелокальными условиями, связывающими значения искомой функции и её производных до третьего порядка (включительно), принимаемых на боковых сторонах прямоугольника. Сначала методом интегралов энергии доказана единственность решения поставленной задачи. Затем, исследована спектральная задача, возникающая при применении метода Фурье, основанном на разделении переменных, к поставленной начально – граничной задаче. Построена функция Грина спектральной задачи, с помощью чего она эквивалентно сведена к интегральному уравнению Фредгольма второго рода с симметричным ядром, откуда следует существование счетного числа собственных значений и собственных функций спектральной задачи. Доказана теорема разложения заданной функции в равномерно сходящийся ряд по системе собственных функций. С помощью найденного интегрального уравнения и теоремы Мерсера доказана равномерная сходимость некоторых билинейных рядов, зависящих от найденных собственных функций. Установлен порядок коэффициентов Фурье. Решение изучаемой задачи выписано в виде суммы ряда Фурье по системе собственных функций спектральной задачи. Исследована равномерная сходимость этого ряда и рядов, полученных из него почленным дифференцированием. Получена оценка для решения задачи, откуда следует его непрерывная зависимость от заданных функций. Recently, initial-boundary problems in a rectangular domain for differential equations in partial derivatives of both even and odd order have been intensively studied. In this case, non-degenerate equations or equations that degenerate on one side of the quadrilateral are taken as the object of study. But initialboundary problems (both local and non-local) for equations with two or three lines of degeneracy remain unexplored. In this paper, in a rectangular domain, a fourth-order equation degene-rating on three sides of the rectangular and contains the Gerasimov-Caputo fractional diffe-rentiation operator has been considered. For this equation, an initial-boundary problem is formulated and investigated, with non-local conditions connecting the values of the desired function and its derivatives up to the third order (inclusive), taken on the sides of the rectangle. From the beginning, the uniqueness of the solution of the formulated problem was proved by the method of energy integrals. Then, the spectral problem that arises when applying the Fourier method based on the separation of variables to the considered initial-boundary problem has been investigated. The Green’s function of the spectral problem was constructed, with the help of which it is equivalently reduced to an integral Fredholm equation of the second kind with a symmetric kernel, which implies the existence of a countable number of eigenvalues and eigenfunctions of the spectral problem. A theorem is proved for expanding a given function into a uniformly convergent series in terms of a system of eigenfunctions. Using the found integral equation and Mercer’s theorem, we prove the uniform convergence of some bilinear series depending on the found eigenfunctions. The order of the Fourier coeffi-cients have been established. The solution of the considered is written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniform convergence of this series and the series obtained from it by term-by-term differentiation is studied. An estimate for solution to problem is obtained, from which follows its continuous dependence on the given functions.

Perturbations of singular fractional SDEs

We obtain well-posedness results for a class of ODE with a singular drift and additive fractional noise, whose right-hand-side involves some bounded variation terms depending on the solution. Examples of such equations are reflected equations, where the solution is constrained to remain in a rectangular domain, as well as so-called perturbed equations, where the dynamics depend on the running extrema of the solution. Our proof is based on combining the Catellier–Gubinelli approach based on Young nonlinear integration, with some Lipschitz estimates in p-variation for maps of Skorokhod type, due to Falkowski and Słominski. An important step requires to prove that fractional Brownian motion, when perturbed by sufficiently regular paths (in the sense of p-variation), retains its regularization properties. This is done by applying a variant of the stochastic sewing lemma.

An effective high-order five-point stencil, based on integrated-RBF approximations, for the first biharmonic equation and its applications in fluid dynamics

PurposeThe purpose of this paper is to present a new discretisation scheme, based on equation-coupled approach and high-order five-point integrated radial basis function (IRBF) approximations, for solving the first biharmonic equation, and its applications in fluid dynamics.Design/methodology/approachThe first biharmonic equation, which can be defined in a rectangular or non-rectangular domain, is replaced by two Poisson equations. The field variables are approximated on overlapping local regions of only five grid points, where the IRBF approximations are constructed to include nodal values of not only the field variables but also their second-order derivatives and higher-order ones along the grid lines. In computing the Dirichlet boundary condition for an intermediate variable, the integration constants are used to incorporate the boundary values of the first-order derivative into the boundary IRBF approximation.FindingsThese proposed IRBF approximations on the stencil and on the boundary enable the boundary values of the derivative to be exactly imposed, and the IRBF solution to be much more accurate and not influenced much by the RBF width. The error is reduced at a rate that is much greater than four. In fluid dynamics applications, the method is able to capture well the structure of steady highly non-linear fluid flows using relatively coarse grids.Originality/valueThe main contribution of this study lies in the development of an effective high-order five-point stencil based on IRBFs for solving the first biharmonic equation in a coupled set of two Poisson equations. A fast rate of convergence (up to 11) is achieved.

Solving the non-local Fokker-Planck equations by deep learning.

Physics-informed neural networks (PiNNs) recently emerged as a powerful solver for a large class of partial differential equations (PDEs) under various initial and boundary conditions. In this paper, we propose trapz-PiNNs, physics-informed neural networks incorporated with a modified trapezoidal rule recently developed for accurately evaluating fractional Laplacian and solve the space-fractional Fokker-Planck equations in 2D and 3D. We describe the modified trapezoidal rule in detail and verify the second-order accuracy. We demonstrate that trapz-PiNNs have high expressive power through predicting the solution with low L 2 relative error by a variety of numerical examples. We also use local metrics, such as point-wise absolute and relative errors, to analyze where it could be further improved. We present an effective method for improving the performance of trapz-PiNN on local metrics, provided that physical observations or high-fidelity simulation of the true solution are available. The trapz-PiNN is able to solve PDEs with fractional Laplacian with arbitrary α ∈ ( 0 , 2 ) and on rectangular domains. It also has the potential to be generalized into higher dimensions or other bounded domains.

Implicit-Explicit Finite Difference Approximations of a Semilinear Heat Equation with Logarithmic Nonlinearity

Abstract We formulate an initial and Dirichlet boundary value problem for a semilinear heat equation with logarithmic nonlinearity over a two-dimensional rectangular domain. We approximate its solution by employing the standard second-order finite difference method for space discretization, and a linearized backward Euler method, or, a linearized BDF2 method for time stepping. For the linearized backward Euler finite difference method, we derive an almost optimal order error estimate in the discrete L t ∞ ⁢ ( L x ∞ ) L^{\infty}_{t}(L^{\infty}_{x}) -norm without imposing mesh conditions, and for the linearized BDF2 finite difference method, we establish an almost optimal order error estimate in the discrete L t ∞ ⁢ ( H x 1 ) L^{\infty}_{t}(H^{1}_{x}) -norm, allowing a mild mesh condition to be satisfied. Finally, we show the efficiency of the numerical methods proposed, by exposing results from numerical experiments. It is the first time in the literature where numerical methods for the approximation of the solution to the heat equation with logarithmic nonlinearity are applied and analysed.

On the solvability of the initial-boundary value problem for a nonlocal hyperbolic equation

In this paper, we consider a partial differential equation with involutively transformed arguments in a rectangular domain. The considered equation is a non-local analog of the second-order hyperbolic type equation. This equation is subject to initial-boundary conditions, and the order of the boundary operators exceeds the order of the equation. Questions of correctness of the considered problem are investigated. To solve the problem, the Fourier method is used, i.e. separation of variables method. The properties of eigenfunctions and eigenvalues of the corresponding spectral problem are studied. For the main problem under consideration, theorems on the uniqueness and existence of a solution are proved. When proving the theorem on the uniqueness of the solution, the problem under study is reduced to two auxiliary, homogeneous initial-boundary value problems for a classical equation of hyperbolic type. The resulting equations depend on the coefficients of the main equation and certain conditions are imposed on them. Further, using the completeness of the eigenfunctions of the auxiliary spectral problem, the solution of the main problem is sought in the form of a series in this system. For the unknown coefficients of the series, a system of ordinary differential equations with high-order boundary conditions is obtained. Solving these problems, we find an explicit form of the solution of the main problem under study.

On one solution of a periodic boundary value problem for a hyperbolic equations

In a rectangular domain, we consider a boundary value problem periodic in one variable for a system of partial differential equations of hyperbolic type. Introducing a new unknown function, this problem is reduced to an equivalent boundary value problem for an ordinary differential equation with an integral condition. Based on the parametrization method, new approaches to finding an approximate solution to an equivalent problem are proposed and its convergence is proved. This made it possible to establish conditions for the existence of a unique solution of a semiperiodic boundary value problem for a system of second-order hyperbolic equations.

Уточненные спектральные свойства задач Дирихле и Неймана для оператора Лапласа в прямоугольной области

В одномерных краевых спектральных задачах размерности собственных подпространств не превосходят некоторого известного числа (как правило 1 или 2). В многомерных самосопряженных задачах с дискретным спектром, несмотря на конечную размерность всех собственных подпространств последовательность кратностей может быть неограничена. Это верно даже для~классических краевых задач, решающихся методом разделения переменных. В случае задач Дирихле и Неймана для оператора Лапласа в прямоугольной области $\Omega=(0;a)\times(0;b)$ хорошо известна явная формула $\lambda_{km} = \big(\frac{\pi k}{a}\big)^2 + \big(\frac{\pi m}{b}\big)^2$ для описания всех собственных значений (индексы $k, m$ принимают положительные или неотрицательные значения соответственно для задачи Дирихле или Неймана). Исследование кратностей сводится к подсчету числа различных упорядоченных пар $(k, m)$, соответствующих одному и тому же числу $\lambda_{km}$. На основе классических и новых результатов теории чисел и теории диофантовых приближений в работе изучаются вопросы взаимного расположения, кратностей и асимптотики собственных значений $\lambda_{km}$ в зависимости от параметров $a$ и $b$. В~случае квадратной области ($a=b$) описан явный алгоритм подсчета кратности любого собственного значения, основанный на разложении натурального числа на простые сомножители и подсчете числа сомножителей вида $4k+1$. Для прямоугольной области установлена зависимость распределения кратностей от того, являются ли числа $f:=a/b$ и $f^2$ рациональными или нет. В~случае $f, f^2 \not\in \mathbb{Q}$ доказано, что все собственные значения однократные, но на сколь угодно близком расстоянии располагается бесконечно много пар собственных значений. На основе уточненной оценки остатка в~проблеме круга Гаусса установлена асимптотическая формула Вейля с~двумя первыми членами и квалифицированной оценкой остатка.

G-Fractional Diffusion on Bounded Domains in Rd

In this paper, we study g-fractional diffusion on bounded domains in Rd with absorbing boundary conditions. A new general and explicit representation of the solution is obtained. We study the first-passage time distribution, showing the dependence on the particular choice of the function g. Then, we specialize the analysis to the interesting case of a rectangular domain. Finally, we briefly discuss the connection of this general theory with the physical application to the so-called fractional Dodson diffusion model, recently discussed in the literature.

Variable stiffness discrete Ritz method for free vibration analysis of plates in arbitrary geometries

Ritz method runs into difficulties when dealing with complex geometry domain due to the difficulty of constructing a global trial function to approximate structural deformation and the complexity of integration. In this paper, a novel numerical approach, discrete Ritz method (DRM), is proposed for the free vibration analysis of arbitrary-shaped plates with arbitrary cutouts. Legendre polynomials are adopted to generate the admissible function. DRM transforms the complex geometry domain into a system of rectangular domain with variable stiffness. Standard energy functionals and computation procedures are established for arbitrary-shaped plates. Then, in terms of Gauss-Legendre quadrature, integration over a complex geometry domain can be numerically simulated by discretizing the rectangular domain using Gauss points, and putting the thickness and stiffness of the Gauss points within the cutouts to be zero. With sufficient Gauss points, the innovative integration strategy can provide high precision integration results, and hence resolve the eigenvalue problem for free vibration of arbitrary-shaped plates with arbitrary cutouts. Numerical examples considering plates with various shapes, cutouts, aspect ratios, and boundary conditions are demonstrated to show the precision and stability of DRM, and results are compared with other available numerical, analytical, and experimental results existing in the literature.