Integral Equation Formulation Research Articles

Fast Computation of Eddy Currents for Multiple Conductors

In this work, the solution to the eddy current problem for slowly moving conductors is investigated with the volume integral formulation of the electric field integral equation (EFIE). On the one hand, the formulation gives the possibility to mesh only the conductors; on the other hand, the inductance matrix is fully populated, thus limiting the size of the problem. Furthermore, the standard solution implies the reassembling of the stiffness matrix for each position assumed by the moving conductors. This results in a limit in the size of the problem to treat and a relevant cost in the computation time. To overcome this limit, in this work, the mutual coupling part of the system with two conductors is computed on the fly, thus giving the possibility to solve problems with a higher number of unknowns. Finally, to speed up the solution of the system, the Gauss–Seidel (GS) iterative techniques and the fast multipole method (FMM) are applied to take into account the mutual effects between the conductors. A comparison of the proposed method with a reference one shows the effectiveness of this technique.

Solving the nonlinear Riccati equation for the 1D plane-wave reflection response from a velocity gradient interface by a Heun to hypergeometric reduction

For 1D acoustic wave propagation in the frequency domain, the complex nonlinear Riccati differential equation describes the dependence of the plane-wave reflection response in depth, defined as the upgoing part of the wavefield divided by the downgoing part of the wavefield at that depth. In the case that the model is a velocity gradient interface expressed in terms of a smooth Heaviside function defined by the Fermi-Dirac function, the Riccati equation is shown to have an analytical solution for the reflection response. The solution accounts for all wave phenomena in the gradient zone. The solution of the Riccati equation for the reflection response is obtained by introducing the Riccati equation for the reciprocal of the reflection response which can be related to a second-order self-adjoint linear differential equation. The two latter equations obey the same radiation condition. Furthermore, a connection between the second-order linear equation and the Heun equation is obtained, both being characterized by having four regular singular points. By a change of variables, the Heun equation can be transformed into Gauss’ hypergeometric equation, whereby the solution is expressed as the sum of two linearly independent functions containing hypergeometric functions. The advantage of applying the Heun-to-hypergeometric reduction to the nonlinear Riccati equation is that solutions to the hypergeometric equation are well known. The two functions allow us to analytically determine the Riccati solution because the ratio of the linearly independent functions at infinity is constant. In contrast to the classic frequency-independent reflection coefficient from two layers in welded contact, the reflection coefficient at the surface associated with the gradient interface depends on frequency. In the limit of zero frequency, the reflection coefficient approaches the classic one. We have implemented the iterative solution of the integral form of the Riccati equation and found that the solution builds up nonlinearly to the correct solution. The model is verified by a numerical example valid for 1D wave propagation.

Development of a fast thermal-hydraulic model to simulate heat and fluid flow in MNSR

Extending a reliable and fast thermal–hydraulic model is the first stage in developing a micro-simulator for a miniature neutron research reactor (MNSR) with training and educational objectives. The framework of the TH model, which comprises an explicit finite difference formulation for solving coupled hydraulic and thermal equations with variable water properties, is explored in this study. The hydraulic section uses the integral form of the momentum equation to relate the mass flow rate variation to buoyancy and friction forces. The thermal part of the model relies on the integral form of the energy equation for different elements of the reactor including fuel, core, reflector, tank, chimney, and pool. The model's results are compared to reactor operation data and available correlations for various reactor powers. According to the comparison, the model's output matches the trend of experimental data for all power in both short and long time periods and predicts the eventual temperature rise of the core with an error of less than 5%. The performance of the model in other operational conditions is also evaluated. The model's performance in different operational circumstances is also assessed, and the results suggest that reducing the reactor's initial temperature would enhance the core's temperature rise.

A scaled boundary finite element formulation for solving plane-strain viscoelastic problems

A new scaled boundary finite element formulation is developed to solve plane-strain viscoelastic problems. In this method, the integral form of linear viscoelasticity constitutive equation and Prony series definition of viscoelastic properties are used to expand scaled boundary finite element formulation. According to the new formulation, an step by step solution process is introduced. The explicit scheme is used for transient analysis, however results are acceptable for either small or large time increments. Several examples with various geometry, material properties, and loading conditions have been solved and the efficiency of the new formulation is verified against FEM. In these examples, elements with different shapes (quadrilateral and polygon), orders (linear and quadratic), and densities are studied. The new formulation also showed promising performance in analyzing nearly incompressible materials. The proposed development will introduce SBFEM as a privileged alternative for FEM in computational biomechanics. The scaled boundary finite element formulation for plane-stress viscoelastic problems is also presented in the appendix.

Decoupled Potential Integral Equation for Electromagnetic Scattering From Arbitrarily Shaped Dielectric Objects

Recently, integral equation formulations that use potentials as opposed to fields as unknown quantities have been developed for scattering from dielectric objects. It has been shown that these formulations can be construed so that they are well-conditioned across a broad frequency spectrum, a result that has been theoretically proven for spherical systems. Unfortunately, to date, this formulation has not been implemented on practical discretizations of objects. This is the goal of this article. Specifically, we present a well-conditioned and well-tested decoupled potential integral equation (DPIE) formulation and all the necessary implementation details for electromagnetic scattering from homogeneous, dielectric, arbitrarily shaped objects. The resulting decoupled systems do not suffer from low-frequency breakdown. Results that demonstrate these properties are presented for a number of different dielectric targets. Furthermore, in order to fully validate each of the two integral equations for the potentials, we develop analytical solutions for spherical systems.

Efficient Partial Elements Computation for the Non-Orthogonal PEEC Method Including Conductive, Dielectrics, and Magnetic Materials

The partial-element equivalent circuit method is a well-known numerical technique that is used to solve Maxwell’s equations in their integral equation form. The application of the partial-element equivalent circuit (PEEC) method to modeling domains with non-orthogonal three-dimensional geometries requires the computation of the interaction integrals to be performed numerically, thus slowing down the overall computation. This work presents a new technique that allows improving the computation of the interaction integrals of the PEEC method for non-orthogonal geometries under the quasi-static hypothesis. To this purpose, a voxelization approach that automatically decomposes non-orthogonal volumes in elementary parallelepipeds is used, allowing the implementation of closed-form formulas for the interaction integrals and completely avoiding numerical integration. The proposed approach is applied to three example problems exhibiting very good accuracy and excellent speed-up compared with the standard one using the numerical integration.

A modified Euler–Maruyama method for Riemann–Liouville stochastic fractional integro-differential equations

In this paper, we propose a modified Euler–Maruyama (EM) method for Riemann–Liouville stochastic fractional integro-differential equations with weakly singular kernels, and then analyse the strong convergence of the proposed EM method. Specifically, we transform the considered stochastic fractional integro-differential equation into its equivalent form of stochastic Volterra integral equations and derive the corresponding modified EM method. Then, the strong convergence order of the proposed method is established, where α is the order of Riemann–Liouville fractional derivative with . Finally, numerical experiments are demonstrated to support our theoretical results.

Energy integral equation for premixed flame-wall interaction in turbulent boundary layers and its application to turbulent burning velocity and wall flux evaluations

An integral form of the energy conservation equation has been derived from the first principle for low Mach number conditions for statistically steady premixed flame-wall interaction within turbulent boundary layers. The validity of this equation has been demonstrated based on three-dimensional Direct Numerical Simulation data of statistically stationary oblique quenching of a turbulent premixed V-shaped flame in a channel flow configuration as a result of its interaction with an inert isothermal wall. It has been found that the wall heat flux and the integral of chemical heat release in the wall normal direction within the turbulent thermal boundary layer are the major contributors in the energy integral equation, and their difference is accounted for by the advection contribution. The magnitudes of the wall heat flux increase, and integral of heat release rate across the thermal boundary layer decrease with increasing distance from the leading edge of the boundary layer as a result of flame quenching. The integral form of the energy conservation equation has been utilised to demonstrate that the Nusselt number (or Stanton number) for wall heat transfer is intrinsically related to the turbulent burning velocity in the case of flame-wall interaction within turbulent boundary layers. A Flame Surface Density based reaction rate closure, modified to account for the near-wall behaviour, has been utilised to estimate the mean Nusselt number in the case of flame-wall interaction within turbulent boundary layers, which revealed that the modelling limitations of the mean reaction rate closure may give rise to inaccuracies in the estimation of the mean Nusselt number. By contrast, the measurements of mean velocity, temperature, and wall heat flux can be utilised to estimate the turbulent burning velocity within the turbulent boundary layer using the newly derived energy integral equation.

A Three-dimensional High-order Numerical Model for the Simulation of the Interaction Between Waves and an Emerged Barrier

We present a new three-dimensional numerical model for the simulation of breaking waves. In the proposed model, the integral contravariant form of the Navier-Stokes equations is expressed in a curvilinear moving coordinate system and are integrated by a predictor-corrector method. In the predictor step of the method, the equations of motion are discretized by a shock-capturing scheme that is based on an original highorder scheme for the reconstruction of the point values of the conserved variables on the faces of the computational grid. On the cell faces, the updating of the point values of the conserved variables is carried out by an exact Riemann solver. The final flow velocity field is obtained by a corrector step which is based exclusively on conserved variables, without the need of calculating an intermediate field of primitive variables. The new three-dimensional model significantly reduces the kinetic energy numerical dissipation introduced by the scheme. The proposed model is validated against experimental tests of breaking waves and is applied to the three-dimensional simulation of the local vortices produced by the interaction between the wave motion and an emerged barrier.

Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation

Abstract In order to explore the fractional differential equations in accounting informatization financial software, the author proposes a system for fractional diffusion wave equations and fractional differential equations, two numerical algorithms with higher precision are given, and the amount of computation is reduced at the same time. First, based on the equivalent integral form of the time fractional diffusion wave equation, using the fractional echelon method and the Crank-Nicolson method, for the time fractional diffusion wave equation, a finite difference scheme is designed, this format has second-order accuracy in both the temporal and spatial directions and is computationally stable. Numerical examples verify the accuracy and effectiveness of this format. Then when dealing with the initial value problem of fractional differential equations with Caputo derivative operator, convert it to the equivalent Voltera integral equation system, an initial approximate solution is obtained by a low-order method, derive the residual and error equations, the idea of applying the stepwise correction of spectral delay correction improves the numerical accuracy of the solution, at the same time, the Richard Askey integral equation is used to reduce the amount of calculation. At last, the high precision and effectiveness of the new method are verified by numerical experiments. Experiments show that: Starting from the equivalent integral form of the fractional diffusion wave equation, a second-order finite-difference scheme of the fractional-order diffusive wave equation is constructed, through numerical experiments, it is verified that the scheme has good accuracy and efficiency. In numerical solution, discrete integral equations have better numerical stability than differential equations, therefore, the format also has better stability. When taking different fractional derivative indices a=1.5 and a=1.8, it can be seen that the difference format constructed by the author, in the time direction, has second-order precision, as expected.

An efficient and accurate chatter prediction method of milling processes with a transition matrix reduction scheme

Efficient and accurate chatter prediction is essential for selection of chatter-free process parameters in order to improve machining productivity and surface quality of the workpiece. This paper proposes a five-point Gaussian quadrature-based chatter prediction method of milling processes together with a transition matrix reduction scheme to improve the computational accuracy and efficiency. The five-point Gauss–Legendre quadrature rule is utilized to approximate the equations of motion of milling systems represented in the form of integral equation. By this means, the convergence rate of the chatter prediction method can be improved. Since the transition matrix describing the relation between the states of current and previous tooth or spindle period is used to evaluate chatter stability, its dimension has a significant effect on the computational efficiency. In this paper, it is theoretically proved that the dimension of the transition matrix constructed by numerical integration-based methods can be reduced by half. Based on this, an efficient scheme is firstly proposed to further decrease the computational time. Typical benchmark examples are employed to verify the proposed method. The results demonstrate that the proposed method with the transition matrix reduction scheme is more efficient and accurate than the existing methods. Meanwhile, experimental validation shows the proposed method can predict the milling chatter accurately.

Computational Simulations of Spray Cooling with Air-Assist Injectors

A new computational procedure for simulating air-assist flat sprays with atomization is described and demonstrated for surface cooling applications. This procedure builds upon and utilizes findings from our previous work; particularly the integral form of the conservation equations which are used to derive explicit quadratic formulas for drop size. These formulations relate the local energetic state to initial drop size produced by primary atomization processes. In this manner, the local liquid and gas phase velocities prior to atomization are used in the quadratic formula to provide the initial drop size and the appropriate local velocities are utilized as the initial droplet momentum state in a discrete particle tracking algorithm. This procedure has been performed and compared to experimental data for drop size and velocity. This furnishes a platform to further study the effects of droplet distributions on heat transfer and momentum transfer between the spray and a heated metal surface. This approach is based on the conservation principles and generalizable, so that it can easily be implemented in any spray geometry for accurate and efficient computations of two-phase flows including spray cooling.

Research on the lubricated characteristics of journal bearing based on finite element method and mixed method

In order to study the lubricated characteristics of journal bearing, a new numerical method including finite element method (FEM) and mixed method is provided in this paper. Based on the FEM and variation principle, the equivalent integral form of static Reynolds equation is proposed and solved accurately by an iso-parametric element with eight nodes. Moreover, the mixed method including dichotomy and secant method is chosen to calculate the equilibrium position of journal bearing. The calculated results obtained by this new model are validated by comparing with those results calculated by previous model including finite difference method (FDM). The maximum margin of relative error is less than 1.6%. Furthermore, the effects of diameter, lubricant viscosity and external load on the equilibrium position and inner pressure are further researched. Besides, the stiffness and damping coefficients of journal bearing is preliminarily calculated. The theoretical model and calculated results can provide useful references for the design of journal bearing and rotor coupled system.

Isogeometric FE-BE method with non-conforming coupling interface for solving elasto-thermoviscoelastic problems

The isogeometric finite element (FE) and boundary element (BE) coupling method with non-conforming interface is applied to solve 3D elasto-thermoviscoelastic problems. The thermoviscoelastic constitutive relation is expressed by Stieltjes integral, and the equivalent integral weak form of the problem is derived according to the principle of virtual work, in which the time integral is discretized by the trapezoidal difference approximation formula. The parameters of viscoelastic material are described by Prony series, and the discrete formula can be simplified by recursive relationship to avoid storing all solutions in time steps. The equilibrium and compatibility conditions of FE and BE subdomains on the coupling interface are used to couple two different system matrices, and the coupling of non-conforming non-uniform rational B-splines (NURBS) surfaces in different subdomains is achieved by using virtual knot insertion technique. In addition, the time-temperature equivalent principle is applied to observe the mechanical behavior of viscoelastic materials over a wide temperature range in a short time. Through several numerical examples, the effects of different working conditions and geometrical parameters of combustion chamber on the structure of the solid rocket motor (SRM) are studied, and the correctness and robustness of this method are verified.

Free transverse vibrations of nanobeams with multiple cracks

A nonlocal model is formulated to study the size-dependent free transverse vibrations of nanobeams with arbitrary numbers of cracks. The effect of the crack is modeled by introducing discontinuities in the slope and transverse displacement at the cracked cross-section, proportional to the bending moment and the shear force transmitted through it. The local compliance of each crack is related to its stress intensity factors assuming that the crack tip stress field is undisturbed (non-interacting cracks).The kinematic field is defined based on the Bernoulli-Euler beam theory, and the small-scale size effect is taken into account by employing the constitutive equation of the stress-driven nonlocal theory of elasticity. In this manner, the curvature at each cross-section is defined as an integral convolution in terms of the bending moments at all the cross-sections and a kernel function which depends on a material characteristic length parameter. The integral form of the nonlocal constitutive equation is elaborated and converted into a differential equation subjected to a set of mathematically consistent boundary and continuity conditions at the nanobeam's ends and the cracked cross-sections. The equation of motion in each segment of the nanobeam between cracks is solved separately and the variationally consistent and constitutive boundary and continuity conditions are imposed to determine the natural frequencies. The model is applied to nanobeams with different boundary conditions and the natural frequencies and the mode shapes are presented at the presence of one to four cracks. The results of the model converge to the experimental results available in the literature for the local cracked beams and to the solutions of the intact nanobeams when the crack length goes to zero. The effects of the crack location, crack length, and nonlocality on the natural frequencies are investigated, also for the higher modes of vibrations. Novel findings including the amplification and shielding effects of the cracks on the natural frequencies are presented and discussed.

A cell-centered finite volume formulation of geometrically exact Simo-Reissner beams with arbitrary initial curvatures.

This article presents a novel total Lagrangian cell‐centered finite volume formulation of geometrically exact beams with arbitrary initial curvatures undergoing large displacements and finite rotations. The choice of rotation parameterization, the mathematical formulation of the beam kinematics, conjugate strain measures, and the linearization of the strong form of governing equations are described. The finite volume based discretization of the computational domain and the governing equations for each computational volume are presented. The discretized integral form of the equilibrium equations is solved using a block‐coupled Newton–Raphson solution procedure. The efficacy of the proposed methodology is presented by comparing the simulated numerical results with classic benchmark test cases available in the literature. The objectivity of strain measures for the current formulation and mesh convergence studies for both initially straight and curved beam configurations are also discussed.

Two-dimensional hydrodynamics of a Janus particle vesicle

We develop a new model, to our knowledge, for the many-body hydrodynamics of amphiphilic Janus particles suspended in a viscous background flow. The Janus particles interact through a hydrophobic attraction potential that leads to self-assembly into bilayer structures. We adopt an efficient integral equation method for solving the screened Laplace equation for hydrophobic attraction and for solving the mobility problem for hydrodynamic interactions. The integral equation formulation accurately captures both interactions for near touched boundaries. Under a linear shear flow, we observe the tank-treading deformation in a two-dimensional vesicle made of Janus particles. The results yield measurements of intermonolayer friction, membrane permeability and, at large shear rates, membrane rupture. The simulation studies include a Janus particle vesicle in both linear and parabolic shear flows, and interactions between two Janus particle vesicles in shear and extensional flows. The hydrodynamics of the Janus particle vesicle is similar to the behaviour of an inextensible, elastic vesicle membrane with permeability.

Method of Moments and T-Matrix Hybrid

Hybrid computational schemes combining the advantages of a method of moments formulation of a field integral equation and T-matrix method are developed in this article. The hybrid methods are particularly efficient when describing the interaction of electrically small complex objects and electrically large objects of canonical shapes such as spherical multilayered bodies where the T-matrix method is reduced to the Mie series making the method an interesting alternative in the design of implantable antennas or exposure evaluations. Method performance is tested on a spherical multilayer model of the human head. Along with the hybrid method, an evaluation of the transition matrix of an arbitrarily shaped object is presented and the characteristic mode decomposition is performed, exhibiting fourfold numerical precision compared to conventional approaches.

A Simplified Discontinuous Galerkin Self-Dual Integral Equation Formulation for Electromagnetic Scattering From Extremely Large IBC Objects

A Simplified Discontinuous Galerkin Self-Dual Integral Equation Formulation for Electromagnetic Scattering From Extremely Large IBC Objects

Smoothed finite element approach for viscoelastic behaviors of general shell structures

This study focuses on the viscoelastic analysis of laminated composite shell structures under long-term creep mechanical loading. An advanced finite element technique named as cell/element-based smoothed discrete shear gap method (CS-DSG3) is employed to obtain the numerical solutions for both elastic and viscoelastic problems owing to its accuracy and rapid convergence. In the CS-DSG3 shell element, each triangular shell element is divided into three DSG3 subtriangles to avoid transverse shear locking. Subsequently, the total element strain is computed from the three partial strains of the subtriangles using a smoothing technique. To overcome the difficulty in the constitutive equations in integral forms for a viscoelastic material, all formulations are transformed into the Laplace domain using the convolution theorem. Finally, time-dependent deformations are obtained and converted back to the real-time domain using inverse Laplace techniques. For validation, various numerical examples of a pinched cylinder, clamped hyperbolic paraboloid, and hemispherical shell formed of isotropic elastic, isotropic viscoelastic, and anisotropic composite materials are selected to investigate creep behavior under mechanical loading. The present study extends the finite element simulation of anisotropic viscoelastic shell structures to achieve high accuracy and efficiency based on the Laplace transform and CS-DSG3.