Meshless method for solving multidimensional radiative transfer in graded index medium

A diffuse approximation meshless (DAM) method is presented to solve the radiative transfer equation (RTE) in a graded index medium. The meshless method can solve the equation directly without using an upwind scheme. Absorbing, emitting and scattering media with different kinds of graded indices in 1D and 2D geometries are tested. Prediction results obtained by the proposed method are compared with different references in order to illustrate the performance of this solution method.     

An IB-LBM implementation for fluid-solid interactions with an MLS approximation for implicit coupling

This paper presents a combination of the implicit immersed boundary (IB) method and the lattice Boltzmann method (LBM) for a fluid-solid interaction involving an immersed body with moving boundaries and complex geometries. In the implicit IB-LBM method, the flow is computed using LBM. The body force caused by the immersed body is not pre-calculated, but implicitly determined using a corrected velocity field that accurately satisfies the no-slip and no-penetration conditions on the interface between the fluid and solid. The information transfer is not using the traditional Dirac delta function and moving-least-square (MLS). But an improved MLS (IMLS) approximation has been developed to combined the implicit IB-LBM method, where the orthogonal function system with a weight function is used as the basis function and overcome these disadvantages of traditional MLS, in which it may cause numerical instabilities because the matrix inversion must be solved and the final equations system is easily ill-conditioned or singular. Several different flow problems (a two-dimensional flow past a static, rotating and oscillating cylinder, and the vortex-induced vibration of a cylinder as well as the free movement of flapping foil) are simulated to validate the accuracy and efficiency of the present implicit IB-LBM method. The simulation results show good agreement with previous numerical and experimental results. It is found that the present IB-LBM is efficient and reliable in dealing with fluid–solid interaction problems.     

A meshless local Galerkin method for the numerical solution of Hammerstein integral equations based on the moving least squares technique

In this paper, a computational scheme is proposed to estimate the solution of one- and two-dimensional Fredholm-Hammerstein integral equations of the second kind. The method approximates the solution using the discrete Galerkin method based on the moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The discrete Galerkin technique for integral equations results from the numerical integration of all integrals in the system corresponding to the Galerkin method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The implication of the scheme for solving two-dimensional integral equations is independent of the geometry of the domain. The new method is simple, efficient and more flexible for most classes of nonlinear integral equations. The error analysis of the method is provided. The convergence accuracy of the new technique is tested over several Hammerstein integral equations and obtained results confirm the theoretical error estimates.     

A meshless method based on the moving least squares (MLS) approximation for the numerical solution of two-dimensional nonlinear integral equations of the second kind on non-rectangular domains

This paper investigates a numerical method for solving two-dimensional nonlinear Fredholm integral equations of the second kind on non-rectangular domains. The scheme utilizes the shape functions of the moving least squares (MLS) approximation constructed on scattered points as a basis in the discrete collocation method. The MLS methodology is an effective technique for approximating unknown functions which involves a locally weighted least square polynomial fitting. The proposed method is meshless, since it does not need any background mesh or cell structures and so it is independent of the geometry of the domain. The scheme reduces the solution of two-dimensional nonlinear integral equations to the solution of nonlinear systems of algebraic equations. The error analysis of the proposed method is provided. The efficiency and accuracy of the new technique are illustrated by several numerical examples.     

Use of geometry in finite element thermal radiation combined with ray tracing

In heat transfer for space applications, the exchanges of energy by radiation play a significant role. In this paper, we present a method which combines the geometrical definition of the model with a finite element mesh. The geometrical representation is advantageous for the radiative component of the thermal problem while the finite element mesh is more adapted to the conductive part. Our method naturally combines these two representations of the model. The geometrical primitives are decomposed into cells. The finite element mesh is then projected onto these cells. This results in a ray tracing acceleration technique. Moreover, the ray tracing can be performed on the exact geometry, which is necessary if specular reflectors are present in the model. We explain how the geometrical method can be used with a finite element formulation in order to solve thermal situation including conduction and radiation. We illustrate the method with the model of a satellite.     

Analytical description of the radiative-conductive heat transfer in a gray medium contained between two diffuse parallel plates

Explicit analytical solutions for the temperature and heat flux in a gray medium contained between two diffuse parallel plates are derived for both pure thermal radiation and coupled conduction-radiation heat transfer. This is achieved by combining the integral equations for the heat flux and temperature predicted by the radiative transfer equation with the corresponding predictions of the discrete ordinates method. The algebraic formulation of this well-known method is used to derive analytical results that agree with their corresponding numerical ones with an accuracy greater than 99.9%, for a large interval of optical thicknesses and conduction-to-radiation factors. The explicit and original solutions, for both pure radiation and radiative-conductive heat transfer, therefore solve the problem of one dimensional steady-state heat transfer in gray cavities.     

Radiative transfer in a two-dimensional rectangular annulus medium

The radiative transfer equation in a two-dimensional rectangular annulus medium is solved numerically. The numerical method is based on a finite difference scheme and a product quadrature discrete-ordinate scheme. The discretized equation of transfer is solved iteratively to give the radiation intensity. The medium is assumed to absorb, emit, and anisotropically scatter radiation. It is exposed to diffusely emitting and diffusely reflecting boundaries. The results of the total intensity for various radiative parameters are presented. The method can be modified easily to solve the rectangular medium without the annulus. Our results in this case compare very well with those of Crosbie et al. [1], Thynell et al. [2], and Wu [3].     

Two-dimensional radiative equilibrium

An exact formulation for the radiative flux and the emissive power is presented for a two-dimensional, finite, planar, absorbing-emitting, gray medium in radiative equilibrium. Exact expressions are obtained for a medium subjected to the following types of boundary conditions: (A) cosine varying collimated radiation, (B) a strip of collimated radiation, (C) cosine varying diffuse radiation, and (D) a uniform temperature strip. The solution for the cosine varying collimated radiation model is used to construct the solutions for the other boundary conditions. The two-dimensional equations are reduced to one-dimensional equations by the method of separation of variables.     

An Eulerian–Lagrangian mixed discrete least squares meshfree method for incompressible multiphase flow problems

Mixed discrete least squares meshfree (MDLSM) method has been developed as a truly meshfree method and successfully used to solve single-phase flow problems. In the MDLSM, a residual functional is minimized in terms of the nodal unknown parameters leading to a set of positive-definite system of algebraic equations. The functional is defined using a least square summation of the residual of the governing partial differential equations and its boundary conditions at all nodal points discretizing the computational domain. Unlike the discrete least squares meshfree (DLSM) which uses an irreducible form of the governing equations, the MDLSM uses a mixed form of the original governing equations allowing for direct calculation of the gradients leading to more accurate computational results. In this study, an Eulerian–Lagrangian MDLSM method is proposed to solve incompressible multiphase flow problems. In the Eulerian step, the MDLSM method is used to solve the governing phase averaged Navier–Stokes equations discretized at fixed nodal points to get the velocity and pressure fields. A Lagrangian based approach is then used to track different flow phases indexed by a set of marker points. The velocities of marker points are calculated by interpolating the velocity of fixed nodal points using a kernel approximation, which are then used to move the marker points as Lagrangian particles to track phases. To avoid unphysical clustering and dispersing of the marker points, as a common drawback of Lagrangian point tracking methods, a new approach is proposed to smooth the distribution of marker points. The hybrid Eulerian and Lagrangian characteristics of the approach used here provides clear advantages for the proposed method. Since the nodal points are static on the Eulerian step, the time-consuming moving least squares (MLS) approximation is implemented only once making the proposed method more efficient than corresponding fully Lagrangian methods. Furthermore, phases can be simply tracked using the Lagrangian phase tracking procedure. Efficiency of the proposed MDLSM multiphase method is evaluated using several benchmark problems and the results are presented and discussed. The results verify the efficiency and accuracy of the proposed method for solving multiphase flow problems.     

A modified discrete-vortex method algorithm with shedding criterion for aerodynamic coefficients prediction at high angle of attack

Low-order methods require less computing power than classical computational fluid dynamics and can be implemented on a laptop computer, which is needed for engineering tasks. Discrete vortex methods are such low order methods that can describe the unsteady separated flow around an airfoil. After a presentation of the leading edge suction parameter discrete vortex method, a modified algorithm is proposed, in order to reduce the computing cost, and compared with the previous one. Several reference unsteady airfoil motions are discussed in terms of gain in the computation time with comparisons between the previous scheme and the present one. The accuracy of the new method is demonstrated through aerodynamic coefficients. The application of the present discrete vortex method to a transient pitching motion of an airfoil is also presented, in order to understand the leading edge vortex formation, and its implication in terms of lift and drag coefficients. The method is not limited to unsteady or transient motions but can also simulate the flow around a constant angle of attack airfoil. In that case, an original method of fast summation of the vortices located far away from the airfoil, allows a linear dependence of the computation time versus the number of vortices shed, which is a great improvement over the quadratic dependence observed in the classical discrete vortex methods. The development of the aerodynamic coefficients with angle of attack, from values ranging between −10° and 90°, is obtained for a purely two-dimensional flow. In particular, the shape of the lift coefficient of the airfoil in the fully detached flow region is established. Comparisons with relevant experimental or computational fluid dynamics data are discussed in order to grasp the influence of upstream turbulence level and three-dimensional effects in the measured data in the fully detached flow region.     

Wavefronts for discrete two-dimensional nonlinear diffusion equations

Existence of wavefronts for discrete two-dimensional evolution models involving diffusive terms with a nonlinear dependence on the discrete ‘gradients’ is studied. For small values of a control parameter, we prove existence of steady wavefronts. We investigate numerically the transition of steady fronts to traveling waves as the control parameter increases.     

Cost prediction for ray shooting in octrees

The ray shooting problem arises in many different contexts and is a bottleneck of ray tracing in computer graphics. Unfortunately, theoretical solutions to the problem are not very practical, while practical methods offer few provable guarantees on performance.

Attempting to combine practicality with theoretical soundness, we show how to provably measure the average performance of any ray-shooting method based on traversing a bounded-degree spatial decomposition, where the average is taken to mean the expectation over a uniform ray distribution. An approximation yields a simple, easy-to-compute cost predictor that estimates the average performance of ray shooting without running the actual algorithm.

We experimentally show that this predictor provides an accurate estimate of the efficiency of executing ray-shooting queries in octree-induced decompositions, irrespective of whether or not the bounded-degree requirement is enforced, and of the criteria used to construct the octrees. We show similar guarantees for decompositions induced by kd-trees and uniform grids. We also confirm that the performance of an octree while ray tracing or running a radio-propagation simulation is accurately captured by our cost predictor, for ray distributions arising from realistic data.     

Two-dimensional isotropic scattering

An exact formulation for the source function, the radiative flux, and the intensity is presented for a two-dimensional, finite, planar, isotropically scattering medium. Exact expressions are obtained for both collimated and diffuse radiation. Cosine varying collimated and diffuse radiation is considered in detail. The solution for the cosine varying collimated radiation model is used to construct the solutions for the other boundary conditions. The two-dimensional integral equations are reduced to one-dimensional equations by the method of separation of variables.     

Simplified radiative models for low-Mach number reactive flows

We introduce a class of numerical models for the simulation of radiative effects in low-Mach number reactive flows. These models are based on simplified P_N approximations for radiative heat transfer, low-Mach asymptotic in the compressible flow, and reduced four-step chemical reaction for reacting species. The models presented here remove on one hand, acoustic wave propagation while retaining the compressibilty effects resulting from combustion and on the other hand, simplify the integro-differential equation for radiative transfer to a set of differential equations independent of the angle variable and compatible to those used for modelling flow and combustion. We briefly discuss the basic discretization methodology for the combined equations and its implementation in a modified projection method. We present validation computations for a two-dimensional methane/air flame in which the computational results are compared for nongray participating media.     

A boundary integral equation method for mode elimination and vibration confinement in thin plates with clamped points

We consider the bi-Laplacian eigenvalue problem for the modes of vibration of a thin elastic plate with a discrete set of clamped points. A high-order boundary integral equation method is developed for efficient numerical determination of these modes in the presence of multiple localized defects for a wide range of two-dimensional geometries. The defects result in eigenfunctions with a weak singularity that is resolved by decomposing the solution as a superposition of Green’s functions plus a smooth regular part. This method is applied to a variety of regular and irregular domains and two key phenomena are observed. First, careful placement of clamping points can entirely eliminate particular eigenvalues and suggests a strategy for manipulating the vibrational characteristics of rigid bodies so that undesirable frequencies are removed. Second, clamping of the plate can result in partitioning of the domain so that vibrational modes are largely confined to certain spatial regions. This numerical method gives a precision tool for tuning the vibrational characteristics of thin elastic plates.     

Determination of radiation reflection and transmission operators in a slab

In the framework of a discrete radiative transfer model, a method is proposed for finding the reflection and transmission operators in the conservative case.     

热载荷作用下平面裂纹问题的奇异积分方程与边界无法

从边界积分方程出发,导出了二维裂纹体热传导问题及热弹性问题的积分方程组,继而使用奇异积分方程与边界元相结合的方法,为其建立了相应的数值求解方法。此外,利用奇异积分方程的主部分析法,严格地证明了裂纹尖端温度梯度场的1/√r 奇异性,并且给出了奇性温度梯度场的精确解。最后。对一些典型例子,做了数值计算。     

Dynamic programming and bicubic spline interpolation

Dynamic programming techniques were used to obtain the spline approximation for a function with prescribed values on the knot points along a line. Extending this procedure to two dimensions, the bicubic spline approximation defined over a two-dimensional region is obtained in this paper employing the methods of dynamic programming. A regular rectangular region as well as a region with irregular boundaries can be handled by this method, avoiding the difficulties of large storage and high dimensionality.     

Solution of the self-adjoint radiative transfer equation on hybrid computer systems

A new technique for simulating three-dimensional radiative energy transfer for the use in the software designed for the predictive simulation of plasma with high energy density on parallel computers is proposed. A highly scalable algorithm that takes into account the angular dependence of the radiation intensity and is free of the ray effect is developed based on the solution of a second-order equation with a self-adjoint operator. A distinctive feature of this algorithm is a preliminary transformation of rotation to eliminate mixed derivatives with respect to the spatial variables, simplify the structure of the difference operator, and accelerate the convergence of the iterative solution of the equation. It is shown that the proposed method correctly reproduces the limiting cases—isotropic radiation and the directed radiation with a δ-shaped angular distribution.     

Half- and finite-range completeness for the equation of transfer of polarized light

On neglecting reflection by the surface the existence and uniqueness are proved for the solution of the equation of transfer of polarized light in a homogeneous semi-infinite or finite plane-parallel medium. A general L_L-space formulation, where 1 ≤ p < ∞, is adopted. The analysis concerns a vector-valued convolution equation, which is an equivalent form of the equation of radiative transfer and is solved with the help of Wiener-Hopf factorization, Fredholm index and cone preservation methods. The results are also proved for the equations obtained from the full equation of transfer by means of Fourier expansion and symmetry relations.