2D Helmholtz Equation Research Articles

Three types of quasi-Trefftz functions for the 3D convected Helmholtz equation: construction and approximation properties

Three types of quasi-Trefftz functions for the 3D convected Helmholtz equation: construction and approximation properties

On the Optimum Distribution of Multiple Helmholtz Resonators for Annular Combustors

Abstract The combustors of many modern land-based gas turbines and aero-engines are annular. This kind of combustors often suffer from thermoacoustic oscillations, with the occurrence of mostly the circumferential first-order, second-order and third-order oscillation modes in the annular combustion chamber. To solve this problem, passive control methods are widely used adding passive acoustic dampers such as Helmholtz resonators (HRs). Depending on the type and number of HRs, and the possible positions over the circumference to install them, there could easily have millions, or even billions, of possible arrangment patterns. Finding a good design is the key to solve this problem. In this paper, we perform a theoretical and numerical study of an annular combustor installed with multiple types of HRs over the circumference. Firstly, a simple annular duct with arbitrary distributions of these HRs is studied analytically by solving the nonlinear eigenvalue problem of a 1D network model. Based on the results of the analytical method, the impact of the HRs on the acoustic modes of the combustion chamber is studied and the optimum arrangement of multiple resonators is obtained. This arrangement usually gives a null (or small) mode splitting strength and a good damping effect. Finally, we apply the optimum arrangement to damp the thermoacoustic modes that have been captured by numerical simulation for a real annular combustor. We use numerical simulations based on solving the 3D Helmholtz equation in COMSOL to verify the feasibility of the optimum arrangement.

Improving the conditioning of the Method of Fundamental Solutions for the Helmholtz equation on domains in polar or elliptic coordinates

A new approach to overcome the ill-conditioning of the Method of Fundamental Solutions (MFS) combining Singular Value Decomposition (SVD) and an adequate change of basis was introduced in [1] as MFS-SVD. The original formulation considered polar coordinates and harmonic polynomials as basis functions and is restricted to the Laplace equation in 2D. In this work, we start by adapting the approach to the Helmholtz equation in 2D and later extending it to elliptic coordinates. As in the Laplace case, the approach in polar coordinates has very good numerical results both in terms of conditioning and accuracy for domains close to a disk but does not perform so well for other domains, such as an eccentric ellipse. We therefore consider the MFS-SVD approach in elliptic coordinates with Mathieu functions as basis functions for the latter. We illustrate the feasibility of the approach by numerical examples in both cases.

A phase velocity preserving fourth-order finite difference scheme for the Helmholtz equation with variable wavenumber

Numerically solving the Helmholtz equation with large wavenumbers can be challenging due to the highly oscillatory nature of the solution. This paper proposes a 3-point finite difference scheme for numerically solving the 1D Helmholtz equation with variable wavenumber. The proposed scheme preserves the phase velocity, making it well-suited for problems with large wavenumbers. Convergence analysis shows that the proposed scheme is guaranteed to have fourth-order accuracy. Numerical results demonstrate that the proposed scheme maintains high accuracy even for problems with large wavenumbers.

Complex analytic dependence on the dielectric permittivity in ENZ materials: The photonic doping example

AbstractMotivated by the physics literature on “photonic doping” of scatterers made from “epsilon‐near‐zero” (ENZ) materials, we consider how the scattering of time‐harmonic TM electromagnetic waves by a cylindrical ENZ region is affected by the presence of a “dopant” in which the dielectric permittivity is not near zero. Mathematically, this reduces to analysis of a 2D Helmholtz equation with a piecewise‐constant, complex valued coefficient a that is nearly infinite (say with ) in . We show (under suitable hypotheses) that the solution u depends analytically on δ near 0, and we give a simple PDE characterization of the terms in its Taylor expansion. For the application to photonic doping, it is the leading‐order corrections in δ that are most interesting: they explain why photonic doping is only mildly affected by the presence of losses, and why it is seen even at frequencies where the dielectric permittivity is merely small. Equally important: our results include a PDE characterization of the leading‐order electric field in the ENZ region as , whereas the existing literature on photonic doping provides only the leading‐order magnetic field.

A scaled boundary finite element method for soil dynamic impedance of pile groups using hybrid quadtree mesh considering horizontal vibration

The numerical modeling of pile-soil dynamic interaction has been a challenging problem in engineering for many years because of the unbounded domain, which typically involves significant computational cost. This paper develops a novel scaled boundary finite element method (SBFEM) that can accurately calculate the soil dynamic impedance of pile groups with end bearing subjected to horizontal vibration. The three-dimensional (3D) problem is first transformed to a two-dimensional (2D) Helmholtz equation in the horizontal plane, exploiting the analytical solution for the vertical modes using separation of variables. The SBFEM is then applied to solve the 2D computational domain, which is divided into bounded and unbounded domains. Based on the SBFEM, an accurate circular artificial boundary condition (ABC) is formulated to simulate the unbounded domain of the soil. A special hybrid quadtree mesh is devised for this problem, where a high-quality structured quadrilateral mesh is generated surrounding the piles. The accuracy of the proposed method is verified using a single pile problem with known analytical solution. The effects of dimensionless frequency, pile numbers, soil depth and relative distance between piles on the soil dynamic impedance are considered.

Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices

Petrov–Galerkin formulations with optimal test functions allow for the stabilization of finite element simulations. In particular, given a discrete trial space, the optimal test space induces a numerical scheme delivering the best approximation in terms of a problem-dependent energy norm. This ideal approach has two shortcomings: first, we need to explicitly know the set of optimal test functions; and second, the optimal test functions may have large supports inducing expensive dense linear systems. A concise proposal on how to overcome these shortcomings has been raised during the last decade by the Discontinuous Petrov–Galerkin (DPG) methodology. However, DPG has also some limitations and difficulties: the method requires ultraweak variational formulations, obtained through a hybridization process, which is not trivial to implement at the discrete level.Our motivation is to offer a simpler alternative for the case of parametric PDEs, which can be used with any variational formulation. Indeed, parametric families of PDEs are an example where it is worth investing some (offline) computational effort to obtain stabilized linear systems that can be solved efficiently in an online stage, for a given range of parameters. Therefore, as a remedy for the first shortcoming, we explicitly compute (offline) a function mapping any PDE parameter, to the matrix of coefficients of optimal test functions (in some basis expansion) associated with that PDE parameter. Next, as a remedy for the second shortcoming, we use the low-rank approximation to hierarchically compress the (non-square) matrix of coefficients of optimal test functions. In order to accelerate this process, we train a neural network to learn a critical bottleneck of the compression algorithm (for a given set of PDE parameters). When solving online the resulting (compressed) Petrov–Galerkin formulation, we employ a GMRES iterative solver with inexpensive matrix–vector multiplications thanks to the low-rank features of the compressed matrix. We perform experiments showing that the full online procedure is as fast as an (unstable) Galerkin approach. We illustrate our findings by means of 2D–3D Eriksson–Johnson problems, together with 2D Helmholtz equation.

BARYCENTRIC RATIONAL INTERPOLATION METHOD OF THE HELMHOLTZ EQUATION WITH IRREGULAR DOMAIN

In the work, a numerical method of the 2D Helmholtz equation with meshless interpolation collocation method is developed, which is defined in arbitrary domain with irregular shape. In our numerical method, based on the Chebyshev points, the partial derivatives and the spatial variables are discretized by the barycentric rational form basis function. After that the differential equations are simplified by employing differential matrix. To verify the the accuracy, effectiveness and stability in our method, some numerical tests based on the three types of different test points are adopted. Moreover, we can also verify that present method can be applied to both variable wave number problems and high wave number problems.

Laplace Residual Power Series Method for Solving Three-Dimensional Fractional Helmholtz Equations

In the present study, the exact solutions of the fractional three-dimensional (3D) Helmholtz equation (FHE) are obtained using the Laplace residual power series method (LRPSM). The fractional derivative is calculated using the Caputo operator. First, we introduce a novel method that combines the Laplace transform tool and the residual power series approach. We specifically give the specifics of how to apply the suggested approach to solve time-fractional nonlinear equations. Second, we use the FHE to evaluate the method’s efficacy and validity. Using 2D and 3D plots of the solutions, the derived and precise solutions are compared, confirming the suggested method’s improved accuracy. The results for nonfractional approximate and accurate solutions, as well as fractional approximation solutions for various fractional orders, are indicated in the tables. The relationship between the derived solutions and the actual solutions to each problem is examined, showing that the solution converges to the actual solution as the number of terms in the series solution of the problems increases. Two examples are shown to demonstrate the effectiveness of the suggested approach in solving various categories of fractional partial differential equations. It is evident from the estimated values that the procedure is precise and simple and that it can therefore be further extended to linear and nonlinear issues.

Sixth-Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

Sixth-Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

An efficient method for 3D Helmholtz equation with complex solution

<abstract><p>The Helmholtz equation as an elliptic partial differential equation possesses many applications in the time-harmonic wave propagation phenomena, such as the acoustic cavity and radiation wave. In this paper, we establish a numerical method based on the orthonormal shifted discrete Chebyshev polynomials for finding complex solution of this equation. The presented method transforms the Helmholtz equation into an algebraic system of equations that can be easily solved. Four practical examples are examined to show the accuracy of the proposed technique.</p></abstract>

Analyzing 3D Helmholtz equations by using the hybrid complex variable element-free Galerkin method

In this study, we present the hybrid complex variable element-free Galerkin (HCVEFG) method for solving 3D Helmholtz equations. The dimension splitting method (DSM) will be introduced into the corresponding governing equation, thus a series of 2D forms can be obtained by splitting the problem domain of 3D Helmholtz equation. For every 2D problem, the shape function can be obtained by using the improved complex variable moving least-squares (ICVMLS) approximation, and the essential boundary condition can be imposed by using the penalty method, thus the discretized equations of 2D problems can be derived by using the corresponding Galerkin weak form. These equations can be coupled by using the finite difference method (FDM) in the dimension splitting direction, thus final formulae of the numerical solution for 3D Helmholtz equation can be obtained. In Sec. 4, the relative errors are given, and the convergence is analyzed numerically. The numerical result of these examples illustrates that the calculation speed can be improved greatly when the HCVEFG method is used rather than the improved element-free Galerkin (IEFG) method.

A Novel BEM–FEM Scheme for the Interaction of Water Waves with Multiple Vertical Cylinders in the Presence of Currents

The interaction of waves and currents with marine structures finds interesting applications, including the study of offshore and shoreline protection systems, as in the case of permeable breakwaters. The latter systems exhibit various benefits, including a decrease in wave run-up, reflected wave energy and load excitation, allowing for the propagation of part of the incident flow to the lee side, facilitating the improvement of water quality in the protected areas. The present work focuses on the modelling and numerical simulation of wave fields, interacting with arrays of vertical cylinders in the presence of currents. The problem is treated in the framework of potential theory in the frequency domain, assuming waves of small steepness, in conjunction with boundary integral formulation. Numerical results are presented and discussed, concerning the structure of the reflected and transmitted 3D flow fields, making the model suitable for optimization purposes; however, it presents increased computational cost. On the other hand, for small current velocities the problem can be approximately considered on the horizontal plane, modelled by the 2D Helmholtz equation with variable coefficients, which is numerically treated by a coupled BEM–FEM scheme. Numerical examples are presented, demonstrating that the latter model is cost-efficient, providing reasonable predictions, and can be used for the preliminary study of the hydrodynamic characteristics of the considered configurations and the support of the design.

High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations

Abstract In this paper, the sixth-order compact finite difference schemes for solving two-dimensional (2D) and three-dimensional (3D) Helmholtz equations are proposed. Firstly, the sixth-order compact difference operators for the second-order derivatives are applied to approximate the Laplace operator. Meanwhile, with the original differential equation, the sixth-order compact difference schemes are proposed. However, the truncation errors of the proposed scheme obviously depend on the unknowns, source function and wavenumber. Thus, we correct the truncation error of the sixth-order compact scheme to obtain an improved sixth-order compact scheme that is more accurate. Theoretically, the convergence and stability of the present improved method are proved. Finally, numerical tests verify that the improved schemes are more accurate.

The Hybrid Element-Free Galerkin Method for 3D Helmholtz Equations

In this study, the hybrid element-free Galerkin (HEFG) method is studied to solve the 3D Helmholtz equations. The idea of the dimension splitting method (DSM) is introduced into the improved element-free Galerkin (IEFG) method, thus a sequence of 2D forms can be obtained by splitting the problem domain of 3D Helmholtz equation, each 2D form can be discretized by using the IEFG method, thus the corresponding 2D discretized equations can be obtained, and these equations can be coupled by employing the finite difference method (FDM) in the dimension splitting direction, thus final formulae of numerical solution for 3D Helmholtz equation can be obtained. In Sec. 4, the relative errors and the convergence are analyzed, respectively, and the numerical results show that the calculation resources can be saved greatly without losing the computational accuracy when using the HEFG method.

A numerical model with high-accuracy and high-efficiency for the soil dynamic impedance of pile groups under vertical vibration

This paper develops a novel numerical model for the calculation of dynamic impedance of soil considering end-bearing pile groups under vertical vibration. The soil layer is treated as a linear viscoelastic material with hysteretic damping. The vertical displacement is assumed to be uniform in each individual pile in the groups at the same embedment depth. Using separation of variables and applying proper boundary conditions at the surface and rigid bedrock, the governing equation of soil layer in three-dimensional (3D) is first transformed into a two-dimensional (2D) Helmholtz equation in xy plane, where the solution in the vertical direction is obtained analytically. To solve this problem efficiently and accurately, the whole 2D unbounded soil is then divided into a finite interior domain and an infinite exterior domain. The finite domain is discretized using the finite element method (FEM) while an exact circular artificial boundary condition (ABC) is developed to model the infinite domain. Finally, the effect of pile group interaction on the dynamic impedance of soil is investigated considering various factors, e.g., dimensionless frequency, pile number, pile slenderness ratio and relative distance between piles.

An analytical method for the solution of earthquake-induced hydrodynamic forces and wave forces on multiple vertical cylinders with arbitrary smooth sections

Analytical methods are developed to investigate the interaction of water with multiple cylinders with arbitrary smooth cross-sections subjected to horizontal ground motion and incident linear waves. Based on the boundary conditions of the seafloor and the water surface, the three-dimensional Laplace equation was transformed into the two-dimensional (2D) Helmholtz equation. Based on the condition that the coefficient function of the cylinder surface can be expanded into a Fourier series, radiation theory and diffraction theory are extended to solve earthquake-induced hydrodynamic forces and wave forces on multiple cylinders. The proposed analytical solution consists of the high-order scattered wave pressures due to diffracted waves. The present method was verified using two double cylinder arrays with elliptical and round-ended cross-sections. Finally, this method was extended to multiple scattering analysis for the interaction of water under earthquake and wave loads.

Sixth-order quasi-compact difference schemes for 2D and 3D Helmholtz equations

Sixth-order quasi-compact difference (QCD) schemes are proposed for the two-dimensional (2D) and the three-dimensional (3D) Helmholtz equations with the variable parameter. Our approach provides the compact mesh stencil for the unknowns, while the noncompact mesh stencil is employed for the source term and the parameter function without involving their derivatives. For the proper interior grid points that are without adjoining the boundary, the sixth-order truncated errors are obtained by the QCD method. Yet the compact scheme is utilized for both of the source term and the parameter function on the improper interior grids that neighbor the boundary, which only reaches the fourth-order local truncated errors. Theoretically, the sixth-order accuracy of the global error by the proposed QCD method is strictly proved for the non-positive constant parameter. Numerical examples are given to demonstrate that the QCD method achieves the global sixth-order convergence for general variable parameters.

Lightning Stokes Solver

Gopal and Trefethen recently introduced “lightning solvers” for the 2D Laplace and Helmholtz equations, based on rational functions with poles exponentially clustered near singular corners. Making use of the Goursat representation in terms of analytic functions, we apply these methods to the biharmonic equation, specifically to 2D Stokes flow. Solutions to model problems are computed to 10-digit accuracy in less than a second of laptop time. As an illustration of the high accuracy, we resolve two or more counterrotating Moffatt eddies near a singular corner.

The Improved Element-Free Galerkin Method for 3D Helmholtz Equations

The improved element-free Galerkin (IEFG) method is proposed in this paper for solving 3D Helmholtz equations. The improved moving least-squares (IMLS) approximation is used to establish the trial function, and the penalty technique is used to enforce the essential boundary conditions. Thus, the final discretized equations of the IEFG method for 3D Helmholtz equations can be derived by using the corresponding Galerkin weak form. The influences of the node distribution, the weight functions, the scale parameters of the influence domain, and the penalty factors on the computational accuracy of the solutions are analyzed, and the numerical results of three examples show that the proposed method in this paper can not only enhance the computational speed of the element-free Galerkin (EFG) method but also eliminate the phenomenon of the singular matrix.