Distribution Of Grid Points Research Articles

Physics Informed Neural Network application on mixing and heat transfer in combined electroosmotic-pressure driven flow

In the current research, different architectures of physics-informed neural networks (PINNs) are implemented to investigate the mixed electroosmotic pressure driven (EOF/PD) flow in microchannels with non-uniform zeta-potential distribution on the walls. Through performing a detailed numerical simulation and PINN solutions based on Poisson-Boltzmann, Laplace, Navier-Stokes, concentration, and energy equations, we predict the non-uniformly distributed zeta-potential on the fluid dynamic, and heat transfer characteristics. It is observed that there is a good agreement between the Finite Volume Method (FVM) and segregated PINN approach. Comparing the PINN results with the numerical simulation, we show that amalgamating losses in all governing equations and applying a single PINN leads to higher training loss when compared to the multi-structured PINN, where a PINN is separately trained for each governing equation. Specifically, the normalized mean absolute percentage errors in the velocity prediction of the single and segregated PINNs are 20.17% and 9.2% respectively. In addition to the structure of the PINN, we also examine the effect of grid point distribution on PINNs. We demonstrate that using boundary layer collocation points can drastically improve the training efficiency and reduce the total loss for EOF/PD flow.

Local Refinement and Adaptive Strategy for a System of Free Boundary Power Options with High Order Compact Differencing

In this research, we propose fourth-order non-uniform Hermitian differencing with a fifth-order adaptive time integration method for pricing system of free boundary exotic power put options consisting of the option value, delta sensitivity, and gamma. The main objective for implementing the above scheme is to carefully account for the irregularity in the locality of the left corner point after fixing the free boundary. Specifically and mainly, we stretch the performance of our proposed method threefold. First, we exploit the non-uniform fourth-order Hermitian scheme to locally concentrate space grid points arbitrarily close to the left boundary. Secondly, we further leverage the adaptive nature of the embedded time integration method, which allows optimal selection of a time step based on the space grid point distribution and regional variation. Thirdly, we introduce a fourth-order combined Hermitian scheme, which requires fewer grid points for computing the near boundary point of the delta sensitivity and gamma. Another novelty is how we approximate the optimal exercise boundary and its derivative using a fifth-order Robin boundary scheme and fourth-order combined Hermitian scheme. Our proposed method consistently achieves reasonable accuracy with very coarse grids and little runtime across the numerical experiments. We further compare the results with existing methods and the ones we obtained from the uniform space grid.

Weighted sparse Bayesian method for direction of arrival estimation based on grid fission

AbstractAn off‐grid weighted sparse Bayesian learning algorithm based on grid fission for direction of arrival estimation is proposed. The existing grid fission algorithms can use fewer grid points with variant intervals to estimate the true DOAs. However, their learning processes are based on the traditional sparse Bayesian algorithm, which only assigns the same prior distribution assumption to the signals on all grids, but ignores the difference of signal distribution of different grid points. It will result in inaccurate fission location and fission direction because of the insufficient resolution of the spatial spectrum, reducing the estimation accuracy. Moreover, the fission strategy will cost much computation time due to the increase of grid points. To solve these problems, the proposed algorithm utilises the orthogonality of signal subspace and noise subspace to design the weights for prior signal distribution assumption, making the peaks of spatial spectrum more pronounced and easy to distinguish, using more accurate estimated DOAs and off‐grid parameter to determine the fission location and direction. In addition, the fission process deletes redundant grid points to simplify calculations. Compared with the existing grid fission algorithms, the proposed method has superior performance in estimation accuracy and computational time.

Evaluation of pollutant exposure using virtual walkers and large-eddy simulation: Application to an idealised urban neighbourhood

Urban air quality studies have primarily focused on pollutant dispersion; however, spatial or temporal concentrations collected at discretely distributed grid points (or fixed receptors) do not reflect the actual pollutant exposure of pedestrians. Using large-eddy simulation (LES) with virtual walkers implemented, this study investigates pollutant exposure of walking agents (or moving receptors) in an urban turbulent boundary-layer flow developed over an aligned building array under the influence of different wind directions. The spatial variability of the exposure risks are found to be better captured by the moving receptors than the fixed receptors along the same agent walking tracks. We demonstrate that the actual exposure can differ significantly from results interpreted from data recorded by the fixed receptors (corresponding to Eulerian estimates) and show that large discrepancies occur in avenues near the source, wherein dispersion of the point release has not occurred on larger spatiotemporal scales. In most scenarios, optimal evacuation routes are shown to be ones that deviate as much as possible from the dominant wind direction; however, one needs to decide the priority of moving to further avenues first or immediately adjusting the walking direction. The results should serve as a useful baseline reference for environmental health impact assessment and evacuation route planning against hazardous releases of air pollutants in more complex urban environments.

A Trajectory Generation Algorithm for a Re-Entry Gliding Vehicle Based on Convex Optimization in the Flight Range Domain and Distributed Grid Points Adjustment

Optimal trajectory generation for the guidance of re-entry glide vehicles is of great significance. To realize a faster generation speed and consistency with the guidance mechanism, an improved convex optimization trajectory generation algorithm based on the flight range domain for the re-entry glide vehicles is proposed in this paper. Firstly, according to the definition of the range-to-go, the projected range-to-go of the re-entry glide vehicle is presented when the dynamic model is converted to the flight range domain. Then, the attack angle and bank angle are expanded to the state variables and the change rate, which is designed as a new control variable. When the dynamic models and constraints are convexificated and discretized, the vehicle trajectory discrete convex model in the flight range domain is proposed. In order to further improve the generation speed and accuracy, an initial trajectory generation method that is close to the guidance requirements is proposed by the landing points of different control laws. In addition, by analyzing the nonlinear illegal degree of grid points, the probability density of grid points and the adjustment strategy of grid points are proposed. Finally, the ablation experiment shows that the initial trajectory generation and distributed grid points method works. With different target points, different no-fly zones, different initial states, and the Monte Carlo experiment, our method can effectively and robustly complete the generation. The lateral and longitudinal generation error is less than 1 km. And compared with the Gaussian pseudo-spectral method, our method obtained comparable accuracy and faster speed.

Construction of grid color mixture model of seven primary-color and modified Stearns-Noechel color matching algorithm for color prediction of full-color-gamut rotor melange yarn

In this paper, the full-color-gamut grid color mixture model containing 601 grid points is constructed by ternary double coupling blending of seven primary-color fibers, and the spinning method of full-color-gamut melange yarn is given by combining with three-channel NC rotor spinning technology. A modified S-N color prediction model was constructed by selecting 55 uniformly distributed grid points for yarn and fabric production from the full-color-gamut grid color mixture model as samples for solving the reflectance conversion coefficients. On this basis, the method of predicting the color value of a melange yarn based on its primary-color fiber composition and blending ratio and predicting the primary-color fiber composition and blending ratio based on the color value of a melange yarn using the parameters of the nearest sample grid point is proposed, and six samples with different blending ratios in six color mixing regions of the full-color-gamut grid color mixture model are designed for validation. The results showed that the average color difference between the predicted color and the actual color of the melange yarn is 1.15, the predicted primary-color fiber composition of the melange yarn is consistent with the actual composition, and the average error between the predicted blending ratio and the actual blending ratio is 3.95%. The method proposed in this paper can effectively predict the color value and blending ratio of melange yarn.

Application of harmonic differential quadrature (HDQ) method for vibration analysis of pipes conveying fluid

Harmonic differential quadrature (HDQ) method is developed to analyze the one dimensional vibration problem of pipes conveying fluid with various boundary conditions. HDQ uses the trigonometric functions to formulate the harmonic test function and weighting coefficients are calculated explicitly. Unevenly spaced grid point distribution method is applied to discretize pipe domain. The high efficiency and accuracy of HDQ method are demonstrated by comparing with other methods for various internal flow induced pipe vibration problems. Numerical simulations show HDQ method holds better computational efficiency and accuracy than other numerical models. HDQ method is more suitable for rapid calculation of large number of numerical cases than other methods.

EquiDistribution Grid Evolution Method for Line Spectrum Estimation

The grid for traditional estimation methods is usually uniform. This is a conservative strategy to form a grid without using any prior information. Estimation results may become better if the true values of parameters are near the artificial grid points. Therefore, a dense grid is usually used to reach expected accuracy, resulting in heavy computation workload. Notice that the performance of estimation is related to the distribution of its grid. Aiming to achieve higher accuracy while use less grid points, a strategy called equidistribution is proposed in this paper. Estimation problem is reformulated by adding the constraint of equidistribution and is solved by a modified grid evolution method, named as EquiDistribution Grid Evolution (EDGE) method. In EDGE, a nonuniform distribution of grid points is evolved according to the distribution of information, and the distribution of information is progressively updated based on the nonuniform grid. The two procedures iterate alternatively. Finally a spectrum based on the adaptively evolved grid is obtained. Compared with previous methods, the proposed method shows advantages on both accuracy and time consumption. The controlling parameter is significantly simplified compared with previous grid evolution method. Furthermore, the resolution and accuracy in parameters’ domain are adaptive and nonuniform. The effectiveness of the EDGE method is demonstrated by numerical simulations.

Performance of Wall-Modeled LES with Boundary-Layer-Conforming Grids for External Aerodynamics

We investigate the error scaling and computational cost of wall-modeled large-eddy simulation (WMLES) for external aerodynamic applications. The NASA Juncture Flow is used as representative of an aircraft with trailing-edge smooth-body separation. Two gridding strategies are examined: 1) constant-size grid, in which the near-wall grid size has a constant value and 2) boundary-layer-conforming grid (BL-conforming grid), in which the grid size varies to accommodate the growth of the boundary-layer thickness. Our results are accompanied by a theoretical analysis of the cost and expected error scaling for the mean pressure coefficient and mean velocity profiles. The prediction of is within less than 5% error for all the grids studied, even when the boundary layers are marginally resolved. The high accuracy in the prediction of is attributed to the outer-layer nature of the mean pressure in attached flows. The errors in the predicted mean velocity profiles exhibit a large variability depending on the location considered, namely, fuselage, wing-body juncture, or separated trailing edge. WMLES performs as expected in regions where the flow resembles a zero-pressure-gradient turbulent boundary layer such as the fuselage ( error). However, there is a decline in accuracy of WMLES predictions of mean velocities in the vicinity of wing-body junctions and, more acutely, in separated zones. The impact of the propagation of errors from the underresolved wing leading edge is also investigated. It is shown that BL-conforming grids enable a higher accuracy in wing-body junctions and separated regions due to the more effective distribution of grid points, which in turn diminishes the streamwise propagation of errors.

Numerical study of grid distribution effect on accuracy of GDQ method of FGPM actuator

The sensors and actuators are made by piezoelectric materials. Some drawbacks are available with the traditional design. To alleviate the drawbacks, Functionally Graded Piezoelectric Materials (FGPMs) have been designed. These materials are widely used in various engineering applications and medical science. Functionally graded piezoelectric actuators play a vital role in medical applications. To transmit a signal, proper bandwidth is required. So the accurate natural frequencies are needed. In this modeling, beam type design is taken into consideration. The available equations are solved by Generalized Differential Quadrature (GDQ) method. The accuracy of GDQ method is sensitive to the grid point distribution. In this work, uniform grid distribution is used to evaluate the natural frequency of FGPM actuator.

ПОСТРОЕНИЕ ОРТОГОНАЛЬНЫХ КРИВОЛИНЕЙНЫХ СЕТОК ДЛЯ РЕГИОНАЛЬНОГО МОДЕЛИРОВАНИЯ ОКЕАНА: МАТЕМАТИЧЕСКИЙ АЛГОРИТМ И РУКОВОДСТВО ПОЛЬЗОВАТЕЛЯ

A new algorithm for constructing orthogonal curvilinear grids on a sphere for a fairly general geometric shape of the modeling region is implemented as a “compile-once - use forever” software package. It is based on the numerical solution of the inverse problem by an iterative procedure -- finding such distribution of grid points along its perimeter, so that the conformal transformation of the perimeter into a rectangle turns this distribution into uniform one. The iterative procedure itself turns out to be multilevel - i.e. an iterative loop built around another, internal iterative procedure. Thereafter, knowing this distribution, the grid nodes inside the region are obtained solving an elliptic problem. It is shown that it was possible to obtain the exact orthogonality of the perimeter at the corners of the grid, to achieve very small, previously unattainable level of orthogonality errors, as well as make it isotropic -- local distances between grid nodes about both directions are equal to each other.

Computational semi-analytical method for the 3D elasticity bending solution of laminated composite and sandwich doubly-curved shells

In this paper, a three-dimensional numerical solution for the bending study of laminated composite doubly-curved shells is presented. The partial differential equations are solved analytically by the Navier summation for the midsurface variables; this method is only valid for shells with constant curvature where boundary conditions are considered simply supported. The partial differential equations present different coefficients, which depend on the thickness coordinates. A semi-analytical solution and the so-called Differential Quadrature Method are used to calculate an approximated derivative of a certain function by a weighted summation of the function evaluated in a certain grin domain. Each layer is discretized by a grid point distribution such as: Chebyshev-Gauss-Lobatto, Legendre, Ding and Uniform. As part of the formulation, the inter-laminar continuity conditions of displacements and transverse shear stresses between the interfaces of two layers are imposed. The proper traction conditions at the top and bottom of the shell due to applied transverse loadings are also considered. The present results are compared with other 3D solutions available in the literature, classical 2D models, Layer-wise models, etc. Comparison of the results show that the present formulation correctly predicts through-the-thickness distributions for stresses and displacements while maintaining a low computational cost.

Modeling acoustic waves in locally enhanced meshes with a staggered-grid finite difference approach

Finite difference is a well-suited technique for modeling acoustic wave propagation in heterogeneous media as well as for imaging and inversion. Typically, the method aims at solving a set of partial differential equations for the unknown pressure field by using a regularly spaced grid. Although finite differences can be fast and cheap to implement, the accuracy of the solution is always restricted by the computational resources. This is a fundamental key point to treat when dealing with large-scale problems. In this work, we present and test a method that uses a non-uniform distribution of grid points to improve on accuracy or to reduce the required computational resources. The applied grid is generated through a coordinate transformation. Differential geometry and generalized coordinates are used to handle and analyze the effect of using a non-uniform grid. Results obtained with the presented method show that the applied transformation as well as the number of points-per-wavelength influences the stability and dispersion in the solution. We exploit this observation to locally improve the accuracy of our simulations. The work presented in this paper allows us to conclude that differential geometry for finite differences can be used to reduce dispersion and hence improve the accuracy when modeling acoustic wave propagation in heterogeneous media. In addition, it can be used to avoid oversampling through the optimization of the number of grid nodes required to have an accurate solution or just honor to the boundaries.

Dynamic modeling and vibration control of a coupled rigid-flexible high-order structural system: A comparative study

A Generalized Differential Quadrature (GDQ) as an accurate numerical technique based on non-uniform grid point distribution, Chebyshev-Gauss-Lobatto (CGL) and Roots of the Legendre Polynomial (RLP) is investigated for active vibration suppression of flexible spacecraft appendages embedded with piezoelectric (PZT) patches. The flexibility of the system is modeled as a sandwich panel with honeycomb core via high-order theories to monitor extra vibrations of the system for high accuracy missions. The coupled governing partial differential equations of the motion and the corresponding boundary conditions were derived through Hamilton's principle. The spacecraft is maneuvered by constant and harmonic torques with different excitation frequency to analyze the vibration sensitivity of the system. The Strain Rate Feedback (SRF) control law is utilized to apply the effects of PZTs action on vibration suppression of flexible appendages. The numerical study of the system characterized by coupled rigid-flexible (high-order) dynamic provides a powerful general tool for analysis of maneuvering spacecraft with smart sandwich appendages and demonstrates the importance of the proposed formulation for the prediction of higher mode vibration response of flexible parts.

An adaptive moving finite element method for steady low Mach number compressible combustion problems

SummaryThis work surveys an r‐adaptive moving mesh finite element method for the numerical solution of premixed laminar flame problems. Since the model of chemically reacting flow involves many different modes with diverse length scales, the computation of such a problem is often extremely time‐consuming. Importantly, to capture the significant characteristics of the flame structure when using detailed chemistry, a much more stringent requirement on the spatial resolution of the interior layers of some intermediate species is necessary. Here, we propose a moving mesh method in which the mesh is obtained from the solution of so‐called moving mesh partial differential equations. Such equations result from the variational formulation of a minimization problem for a given target functional that characterizes the inherent difficulty in the numerical approximation of the underlying physical equations. Adaptive mesh movement has emerged as an area of intense research in mesh adaptation in the last decade. With this approach, points are only allowed to be shifted in space leaving the topology of the grid unchanged. In contrast to methods with local refinement, data structure hence is unchanged and load balancing is not an issue as grid points remain on the processor where they are. We will demonstrate the high potential of moving mesh methods for effectively optimizing the distribution of grid points to reach the required resolution for chemically reacting flows with extremely thin boundary layers.

A Particle Swarm Optimization-Based Method for Numerically Solving Ordinary Differential Equations

The Euler method is a typical one for numerically solving initial value problems of ordinary differential equations. Particle swarm optimization (PSO) is an efficient algorithm for obtaining the optimal solution of a nonlinear optimization problem. In this study, a PSO-based Euler-type method is proposed to solve the initial value problem of ordinary differential equations. In the typical Euler method, the equidistant grid points are always used to obtain the approximate solution. The existing shortcoming is that when the iteration number is increasing, the approximate solution could be greatly away from the exact one. Here, it is considered that the distribution of the grid nodes could affect the approximate solution of differential equations on the discrete points. The adopted grid points are assumed to be free and nonequidistant. An optimization problem is constructed and solved by particle swarm optimization (PSO) to determine the distribution of grid points. Through numerical computations, some comparisons are offered to reveal that the proposed method has great advantages and can overcome the existing shortcoming of the typical Euler formulae.

Solution of Two Dimensional Poisson Equation Using Finite Difference Method with Uniform and Non-uniform Mesh Size

This study focus on the finite difference approximation of two dimensional Poisson equation with uniform and non-uniform mesh size. The Poisson equation with uniform and non-uniform mesh size is a very powerful tool for modeling the behavior of electro-static systems, but unfortunately may not be solved analytically for very simplified models. Consequently, numerical simulation must be utilized in order to model the behavior of complex geometries with practical value. In most engineering problems are also coming from steady reaction-diffusion and heat transfer equation, in elasticity, fluid mechanics, electrostatics etc. the solution of meshing grid is non-uniform and uniform where fine grid is identified at the sensitive area of the simulation and coarse grid at the normal area.The discretization of non-uniform grid is done using Taylor expansion series. The purpose of such discretization is to transform the calculus problem to numerical form (as discrete equation). Therefore, in this study the two dimensional Poisson equation is discretazi with uniform and non-uniform mesh size using finite difference method for the comparison purpose. More over we also examine the ways that the two dimensional Poisson equation can be approximated by finite difference over non-uniform meshes, As result we obtain that for uniformly distributed gird point the finite difference method is very simple and sufficiently stable and converge to the exact solution whereas in non-uniformly distributed grid point the finite difference method is less stable, convergent and time consuming than the uniformly distributed grid points. Keywords: Finite difference method, two dimensional Poisson equations, Uniform mesh size, Non-uniform mesh size, Convergence, Stability, Consistence. DOI : 10.7176/APTA/79-01 Publication date :September 30 th 2019

Analysis and Realization of Sampling Strategy in Coordinate Metrology

The stringent customer demands and competitive market emphasize the importance of efficient and effective inspection in industrial metrology. Therefore, the implementation of an appropriate sampling strategy, i.e., the number of points and their distribution, has become very important in the inspection process using a coordinate measuring machine. Moreover, the quality of inspection results has frequently been influenced by sampling plan as well as workpiece size and surface characteristics. It has been an indispensable problem in the present-day measurement processes. Thus, this paper investigates various sample sizes and different point distribution algorithms that can be employed in the evaluation of form error. The effect of specimen size and surface quality on the sampling strategy has also been investigated. Furthermore, this work employs a fuzzy based Technique for Order Performance by Similarity to Ideal Solution approach to realize the best sampling strategy. The results have demonstrated the significance of robust optimization techniques as well as the importance of a suitable sampling strategy in coordinate metrology. This study has also established that Poisson point distribution achieved the best accuracy and the Grid point distribution had taken the least measurement time.

Order-Preserving Interpolation for Summation-by-Parts Operators at Nonconforming Grid Interfaces

We study non-conforming grid interfaces for summation-by-parts finite difference methods applied to partial differential equations with second derivatives in space. To maintain energy stability, previous efforts have been forced to accept a reduction of the global convergence rate by one order, due to large truncation errors at the non-conforming interface. We avoid the order reduction by generalizing the interface treatment and introducing order preserving interpolation operators. We prove that, given two diagonal-norm summation-by-parts schemes, order preserving interpolation operators with the necessary properties are guaranteed to exist, regardless of the grid-point distributions along the interface. The new methods retain the stability and global accuracy properties of the underlying schemes for conforming interfaces.

Numerical Simulation of Unsteady Capillary-Gravity Waves

In this work, plane problems of numerical simulation of wave motion are studied. Potential flows of a perfect incompressible fluid are considered. A numerical algorithm for calculating the shape of a free boundary is proposed. The algorithm is based on the boundary element method with the use of quadrature formulas with no saturation. The algorithm is used for studying the breaking of capillary gravity waves and calculating thin cumulative jets. The stability of the scheme and high accuracy in calculations of sharp cumulative jets are achieved due to special control for the distribution of grid points and a decrease in the grid step in the neighborhood of the forward end of the cumulative jet with an ultimately rapid growth of the curvature.