Adaptive Grid Method Research Articles

Laser Ignition Process of Energetic Particles Under Consideration of Non‐Fourier Effect

AbstractThe heating and ignition process of energetic material particles under the action of the laser is investigated using numerical simulation. Due to the two characteristics of the high energy density of the laser and the high transients of the whole process, an accurate description is required with the help of the non‐Fourier heat transfer theory. The corresponding mathematical model in the form of partial differential equations is developed for the laser action on the spherical particles. In the process of numerically solving the partial differential equation, adaptive grid method based on wavelets is introduced to improve the finite difference method. Based on the comparison of ignition delay time, the results obtained from numerical calculations are in agreement with the data from experimental tests.

A multiresolution adaptive grid method for solving the generalized probability density equation in stochastic dynamic analysis

A multiresolution adaptive grid method for solving the generalized probability density equation in stochastic dynamic analysis

Numerical Simulation of Water Entry and Exit Problems by an Adaptive Cartesian Grid Method

Numerical Simulation of Water Entry and Exit Problems by an Adaptive Cartesian Grid Method

Optimization of Agricultural Machinery Task Scheduling Algorithm Based on Multiobjective Optimization

In order to improve the effect of agricultural machinery task scheduling, this paper starts from the perspective of multiobjective optimization to achieve task scheduling based on multiobjective particle swarm optimization algorithm. The position of each particle is a combination of resource options for each construction activity, and the displacement range and speed range of the particles are determined. Moreover, this paper uses the method of introducing an external storage library to store the current noninferior solutions and uses the adaptive grid method and the roulette selection method to select the global optimal solution of the particles. In addition, this paper proposes a task scheduling algorithm suitable for modern agricultural machinery based on the actual needs of current agricultural machinery task scheduling. The experimental results show that the agricultural machinery task scheduling algorithm based on multiobjective optimization proposed in this paper has a good agricultural machinery task scheduling effect and meets the basic purpose of optimizing the algorithm in this paper.

A K-means Optimization Algorithm Suitable for Fast Clustering of WebGIS Massive Data

K-means has the advantage of fast speed and is suitable for clustering large-scale data of WebGIS geographic information. However, due to the random selection of K-means initial clustering centers, the clustering results are unstable and the clustering accuracy is poor. Some current research documents have solved the problems of clustering accuracy and stability, but the clustering time has been greatly increased. The article proposes a grid-based K-means improved algorithm GBK-means, which is based on an adaptive grid method to obtain initial clustering centers. Firstly, the parameters of the grid division are obtained by judging the distribution state of the sample data; then, the interconnected areas of each dense grid are obtained and the cluster centers are obtained, and on this basis, the initial cluster centers are obtained. The experimental results on the real data set of WebGIS show that GBK-means has better clustering effect and faster clustering speed than K-means, K-means++, literature [2], and literature [3]. The average value of its F value, accuracy rate and adjusted Rand coefficient (ARI) is 10.9%, 11% and 11.2% higher than that of K-means. The average clustering time is 75.4%, 52.3%, 85.1%, 91.1% faster than K-means, K-means++, literature [2], and literature [3].

A Novel Second-Order Adaptive Grid Method for Singularly Perturbed Convection–Diffusion Equations

In this paper, a singularly perturbed convection–diffusion equation is studied. At first, the original problem is transformed into a parameterized singularly perturbed Volterra integro-differential equation by using an integral transform. Then, a second-order finite difference method on an arbitrary mesh is given. The stability and local truncation error estimates of the discrete schemes are analyzed. Based on the mesh equidistribution principle and local truncation error estimation, an adaptive grid algorithm is given. In addition, in order to calculate the parameters of the transformation equation, a nonlinear unconstrained optimization problem is constructed. Numerical experiments are given to illustrate the effectiveness of our presented adaptive grid algorithm.

Numerical analysis of a system of semilinear singularly perturbed first‐order differential equations on an adaptive grid

An adaptive grid method based on the backward Euler formula for a system of semilinear singularly perturbed initial value problems is studied. Based on the a priori error analysis and mesh equidistribution principle, we prove that the convergence rate of our semidiscrete adaptive grid method is first order, which is robust with respect to the perturbation parameters. Then, in order to construct a fully discrete adaptive grid method, a standard residual‐type a posterior error estimation is constructed by using the linear polynomial interpolation technique. A partly heuristic argument based on this a posteriori error estimator leads to an optimal monitor function, which is used to design an adaptive grid algorithm. Furthermore, we also extend our presented adaptive grid method to a nonlinear system of singularly perturbed problem arising in the modeling of enzyme kinetics and a system of singularly perturbed delay differential equations, respectively. Finally, numerical results are provided to illustrate the effectiveness of our presented adaptive grid method.

Mixed Mesh Finite Volume Method for 1D Hyperbolic Systems with Application to Plug-Flow Heat Exchangers

We present a finite volume method formulated on a mixed Eulerian-Lagrangian mesh for highly advective 1D hyperbolic systems altogether with its application to plug-flow heat exchanger modeling/simulation. Advection of sharp moving fronts is an important problem in fluid dynamics, and even a simple transport equation cannot be solved precisely by having a finite number of nodes/elements/volumes. Finite volume methods are known to introduce numerical diffusion, and there exist a wide variety of schemes to minimize its occurrence; the most recent being adaptive grid methods such as moving mesh methods or adaptive mesh refinement methods. We present a solution method for a class of hyperbolic systems with one nonzero time-dependent characteristic velocity. This property allows us to rigorously define a finite volume method on a grid that is continuously moving by the characteristic velocity (Lagrangian grid) along a static Eulerian grid. The advective flux of the flowing field is, by this approach, removed from cell-to-cell interactions, and the ability to advect sharp fronts is therefore enhanced. The price to pay is a fixed velocity-dependent time sampling and a time delay in the solution. For these reasons, the method is best suited for systems with a dominating advection component. We illustrate the method’s properties on an illustrative advection-decay equation example and a 1D plug flow heat exchanger. Such heat exchanger model can then serve as a convection-accurate dynamic model in estimation and control algorithms for which it was developed.

Numerical study on the influence of liquid viscosity ratio on the hydrodynamics of a single bubble in shear-thinning liquid

The volume of fluid method combined with the adaptive grid method is used to study the influence of shear-thinning liquid viscosity ratio (ε) for various values of rheological index (n) and characteristic time (λ) on bubble shape, rise velocity, and apparent viscosity around a bubble; where 1/λ represents the shear rate at which the change in the viscosity curve occurs from the low shear rate to the power-law region. The results show that, depending on the magnitudes of n and λ, the bubble shape and deformation parameter are strongly related to ε, and bubble shape can exhibit both stable and unstable states. There are two types of deformation phenomena for the unsteady deformation of bubbles. First, the aspect ratio changes with time, whereas the shape remains similar throughout. Second, the shape changes periodically with time. For the given values of n and λ, the bubble rise velocity at large time shows a nonlinear increase as ε increases from 10 to ∞. Under the conditions of n ≤ 0.3, λ* ≥ 68.6, and ε ≥ 100 (λ* is the dimensionless characteristic time), a viscosity blind region (a region of high viscosity, where ε is close to 1) appears behind the bubble. The tendency for the viscosity blind region is to reduce the bubble rise velocity at large time.

A wide-template and high-accuracy data transfer method for unstructured adjoint-based grid adaptation

Grid adaptation is an important way to improve the identification of flow field and the accuracy of aerodynamic characteristics. Adjoint-based error estimation and grid adaptation method focuses on improving aerodynamic characteristics by optimizing local grids areas which are more sensitive to output functions. The data transfer between different meshes during the iteration of adjoint-based grid adaptation is the basis of accurate error estimation and grid adaptation. The traditional vertex-based method, however, is not suitable for cell-centred scheme finite volume method. A new wide-template and high-accuracy data transfer method for unstructured adjoint-based grid adaptation is proposed in this paper. Numerical results show that the proposed method is more accurate than traditional node-based method when transferring discontinuous flow field containing shock waves in cell-centred scheme finite volume method, and the accuracy of aerodynamic characteristics is obviously improved after grid adaptation.

Richardson Extrapolation Method on an Adaptive Grid for Singularly Perturbed Volterra Integro-Differential Equations

In this paper, an adaptive grid method for singularly perturbed Volterra integro-differential equations exhibiting a boundary layer is studied. At first, based on an arbitrary nonuniform mesh, we use the classical backward Euler formula to discrete the first-order derivative part and the left rectangle formula to approximate the integral part, respectively. Meanwhile, it is shown from the mesh equidistribution principle and the truncation error analysis that the presented adaptive grid method is first-order uniformly convergent with respect to the perturbation parameter. Furthermore, the Richardson extrapolation technique is utilized to improve the uniform accuracy of the presented adaptive grid method in the discrete supremum norm form to where is the mesh parameter. Finally, some numerical results are given to support the theoretical results.

A posteriori error estimation and adaptive strategy for a nonlinear fractional differential equation

A nonlinear fractional differential equation with Caputo fractional derivative is considered. The problem is discretized by an upwind finite difference scheme for which a posteriori error analysis in the maximum norm is derived. A partly heuristic argument based on this a posteriori error estimation leads to several suitable monitor functions, and a new monitor function is constructed to design an adaptive grid algorithm. Numerical results are presented to illustrate the performance of the presented adaptive method. Compared with the other monitor functions, the presented adaptive grid method based on the new monitor function is more suitable to solve this type of nonlinearfractional differential equation.

A practical adaptive grid method for the Allen–Cahn equation

We present a simple and practical adaptive finite difference method for the Allen–Cahn (AC) equation in the two-dimensional space. We use a temporally adaptive narrow band domain embedded in the uniform discrete rectangular domain. The narrow band domain is defined as a neighboring region of the interface. We employ a recently developed explicit hybrid numerical scheme for the AC equation. Therefore, the computational algorithm on the narrow band discrete domain is simple and fast. We demonstrate the high performance of the proposed adaptive method for the AC equation through various computational experiments.

An efficient adaptive grid method for a system of singularly perturbed convection-diffusion problems with Robin boundary conditions

A system of singularly perturbed convection-diffusion equations with Robin boundary conditions is considered on the interval [0,1]. It is shown that any solution of such a problem can be expressed to a system of first-order singularly perturbed initial value problem, which is discretized by the backward Euler formula on an arbitrary nonuniform mesh. An a posteriori error estimation in maximum norm is derived to design an adaptive grid generation algorithm. Besides, in order to establish the initial values of the original problems, we construct a nonlinear optimization problem, which is solved by the Nelder–Mead simplex method. Numerical results are given to demonstrate the performance of the presented method.

奇异摄动反应扩散方程的后验误差估计及自适应算法

An adaptive grid method for singularly perturbed semilinear reactiondiffusion equations was studied. The equation was discretized by means of the upwind finite difference scheme on an arbitrary nonuniform mesh. A posteriori error estimation of the presented numerical scheme was built. In turn, the posteriori error bound was derived, and the adaptive grid generation algorithm was designed. Numerical experiments prove the effectiveness of the proposed adaptive grid method.

Laser Ignition Process Subject to Non-Fourier Effect Solved Using Adaptive Grid Method Based on Second-Generation Wavelets

Laser Ignition Process Subject to Non-Fourier Effect Solved Using Adaptive Grid Method Based on Second-Generation Wavelets

An Algorithm for Nonparametric Estimation of a Multivariate Mixing Distribution with Applications to Population Pharmacokinetics.

Population pharmacokinetic (PK) modeling has become a cornerstone of drug development and optimal patient dosing. This approach offers great benefits for datasets with sparse sampling, such as in pediatric patients, and can describe between-patient variability. While most current algorithms assume normal or log-normal distributions for PK parameters, we present a mathematically consistent nonparametric maximum likelihood (NPML) method for estimating multivariate mixing distributions without any assumption about the shape of the distribution. This approach can handle distributions with any shape for all PK parameters. It is shown in convexity theory that the NPML estimator is discrete, meaning that it has finite number of points with nonzero probability. In fact, there are at most N points where N is the number of observed subjects. The original infinite NPML problem then becomes the finite dimensional problem of finding the location and probability of the support points. In the simplest case, each point essentially represents the set of PK parameters for one patient. The probability of the points is found by a primal-dual interior-point method; the location of the support points is found by an adaptive grid method. Our method is able to handle high-dimensional and complex multivariate mixture models. An important application is discussed for the problem of population pharmacokinetics and a nontrivial example is treated. Our algorithm has been successfully applied in hundreds of published pharmacometric studies. In addition to population pharmacokinetics, this research also applies to empirical Bayes estimation and many other areas of applied mathematics. Thereby, this approach presents an important addition to the pharmacometric toolbox for drug development and optimal patient dosing.

An RBF-Based h-Adaptive Cartesian Grid Refinement Method for Arbitrary Single/Multi-Body Hull Modeling and Reconstruction

Complex single/multi-body structures are generally found in ship and ocean engineering. They have the smooth, sharp, concave, and convex surface features in common. Precise modeling of the structures is the basis of numerical simulation. However, the most widely used explicit modeling method requires considerable manual operations. The result is also difficult to reproduce. Therefore, this paper presents a Radial basis function (RBF) based hierarchical (h-) adaptive Cartesian grid method. The RBF is introduced for arbitrary implicit modeling over the Cartesian framework. Thanks to its natural properties, the RBF is easy to use, highly automated, and only needs a set of scatter points for modeling. The block-based h-adaptive mesh refinement (AMR) combined with the RBF aims to enhance the local grid resolution. It accelerates the calculation of intersecting points compared with the uniform Cartesian grid. The accuracy, efficiency, and robustness of the proposed method are validated by the simulation of the 3D analytical ellipsoidal surface and the unclosed conic surface. To select suitable parameters, we thoroughly investigated the uncertainty factors including sample points, RBF functions, and h-AMR grids. The simulation results of the single/multi-body Wigley hull and KCS hull forms verified the proper selection of the factors and the feasibility of our method to solve practical problems.

A fully adaptive nonintrusive reduced-order modelling approach for parametrized time-dependent problems

We present an approach to build a reduced-order model for nonlinear, time-dependent, parametrized partial differential equations in a nonintrusive manner. The approach is based on combining proper orthogonal decomposition (POD) with a Smolyak hierarchical interpolation model for the POD coefficients. The sampling of the high-fidelity model to generate the snapshots is based on a locally adaptive sparse grid method. The novelty of the work is in the adaptive sampling of time, which is treated as an additional parameter. The goal is to have a robust and efficient sampling strategy that minimizes the risk of overlooking important dynamics of the system while disregarding snapshots at times when the dynamics are not contributing to the construction of the reduced model. The developed algorithm was tested on three numerical tests. The first was an advection problem parametrized with a five-dimensional space. The second was a lid-driven cavity test, and the last was a neutron diffusion problem in a subcritical nuclear reactor with 11 parameters. In all tests, the algorithm was able to detect and include more snapshots in important transient windows, which produced accurate and efficient representations of the high-fidelity models.

A second-order adaptive grid method for a nonlinear singularly perturbed problem with an integral boundary condition

In this paper, a nonlinear singularly perturbed problem with an integral boundary condition is studied. A hybrid finite difference method based on the midpoint difference scheme and the upwind difference scheme is constructed. An adaptive grid generation algorithm is generated by equidistributing a monitor function. The convergence analysis of the hybrid difference method on the adaptive grid is derived. It is proved by theory and experiments that the scheme is second-order uniformly convergent, which improves previous results.