Interpolating Moving Least-squares Research Articles

A semiclassical study of cis-trans isomerization in HONO using an interpolating moving least-squares potential.

The interpolating moving least-squares (IMLS) approach for constructing potential energy surfaces has been developed and employed in standard classical trajectory simulations in the past few years. We extend the approach to the tunneling regime by combining the IMLS fitting method and the semiclassical scheme that incorporates tunneling into classical trajectory calculations. Dynamics of cis-trans isomerization in nitrous acid (HONO) is studied as a test case to investigate various aspects of the approach such as the strategy for growing the surface, the basis set employed, the scaling of the IMLS fits, and the accuracy of the surface required for obtaining converged rate coefficients. The validity of the approach is demonstrated through comparison with other semiclassical and quantum mechanical studies on HONO.

Visual simulation of turbulent fluids using MLS interpolation profiles

A detailed description of turbulent fluids based on numerical simulation is an important research topic required by many visual effects. We propose a novel method to simulate fluids with turbulent small-scale details. By inserting diffusive derivatives and divergence-free constraints to moving least-squares (MLS) fitting, we upgrade the velocity interpolation method for existing fluid solvers to enhance the subgrid accuracy. The time-step restriction of asymptotic property of diffusive derivatives is resolved by means of coupling to the constrained interpolation profile (CIP) advection framework. The proposed constrained moving least-squares interpolation profile (CMIP) method provides intuitive control over turbulence through the adjustment of one parameter as though controlling the Reynolds number with an inviscid model. The proposed method generates improved visuals of the highly turbulent fluid and is complementary to existing techniques that are currently being used.

THE INTERPOLATING ELEMENT-FREE GALERKIN (IEFG) METHOD FOR TWO-DIMENSIONAL ELASTICITY PROBLEMS

In this paper, a new method for deriving the moving least-squares (MLS) approximation is presented, and the interpolating moving least-squares (IMLS) method proposed by Lancaster is improved. Compared with the IMLS method proposed by Lancaster, a simpler formula of the shape function is given in the improved IMLS method in this paper so that the new method has higher computing efficiency. Combining the shape function constructed by the improved IMLS method with Galerkin weak form of the elasticity problems, the interpolating element-free Galerkin (IEFG) method for the two-dimensional elasticity problems is presented, and the corresponding formulae are obtained. In the IEFG method, the boundary conditions can be applied directly which makes the computing efficiency higher than the conventional EFG method. Some numerical examples are presented to demonstrate the validity of the method.

The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems

In this paper, the moving least-squares (MLS) approximation and the interpolating moving least-squares (IMLS) method proposed by Lancaster are discussed first. A new method for deriving the MLS approximation is presented, and the IMLS method is improved. Compared with the IMLS method proposed by Lancaster, the shape function of the improved IMLS method in this paper is simpler so that the new method has higher computing efficiency. Then combining the shape function of the improved IMLS method with Galerkin weak form of the potential problem, the interpolating element-free Galerkin (IEFG) method for the two- dimensional potential problem is presented, and the corresponding formulae are obtained. Compared with the conventional element-free Galerkin (EFG) method, the boundary conditions can be applied directly in the IEFG method, which makes the computing efficiency higher. For the purposes of demonstration, some selected numerical examples are solved using the IEFG method.

An Improved Element-Free Galerkin Method for a Kind of KdV Equations

An improved element-free Galerkin method is presented for the numerical solution of the third-order nonlinear KdV equation by coupling the interpolating moving least-squares (IMLS) method with the Galerkin method. The shape function of the IMLS method satisfies the property of Kronecker Delta function, and then the essential boundary condition can be applied directly and easily without any additional numerical effort. A variational method is used to obtain the discrete equations. A numerical example is given to demonstrate the effectiveness of the method presented in this paper for KdV equation.

A Simplified Interpolating Moving Least-Squares Method and its Error Estimates

A disadvantage of the MLS approximation is that the shape function of this method does not satisfy the property of Kronecker Delta function. Thus developing an interpolating MLS approximation is very important. In this paper, the interpolating moving least-squares (IMLS) method presented by Lancaster and Salkauskas is discussed in detail and a simplified expression of the approximation function of the IMLS method is given. The simpler expression makes it more convenient to use this method. The error estimate of the approximation function also is discussed. And a numerical example is given to confirm the results.

A Meshless Method Based on the Improved Interpolating Moving Least-Squares Method for the Regularized Long Wave Equation

This paper presents an improved interpolating moving least-squares (IIMLS) method, in which orthogonal functions system is used as the basis functions. In the IIMLS method, the final algebra equation system is not ill-conditioned, and can be solved without obtaining the inverse matrix. Hence, the computing speed and efficiency are improved. Then based on the IIMLS method, a meshless method is presented for the numerical solution of the regularized long wave (RLW) equation, which can be used to describe phenomena with weak nonlinearity and dispersion waves. And a numerical example is given to confirm the IMLS method.

The Case Study Analysis of Strain–Stress for Jointed Rock Mass Using Element Free Galerkin Method and Paton’s Failure

In this paper evaluated a case study for jointed rock mass using element free Galerkin method and Paton’s failure. The displacement field of each block is constructed by the moving least-squares interpolation with an array of points distributed in the block. The essential boundary conditions are enforced using penalty method. It is shown that excellent results are obtained and less calculation time is required as compared to finite difference method.

Nitrous oxide dimer: A new potential energy surface and rovibrational spectrum of the nonpolar isomer

The spectrum of nitrous oxide dimer was investigated by constructing new potential energy surfaces using coupled-cluster theory and solving the rovibrational Schrödinger equation with a Lanczos algorithm. Two four-dimensional (rigid monomer) global ab initio potential energy surfaces (PESs) were made using an interpolating moving least-squares (IMLS) fitting procedure specialized to describe the interaction of two linear fragments. The first exploratory fit was made from 1646 CCSD(T)/3ZaP energies. Isomeric minima and connecting transition structures were located on the fitted surface, and the energies of those geometries were benchmarked using complete basis set (CBS) extrapolations, counterpoise (CP) corrections, and explicitly correlated (F12b) methods. At the geometries tested, the explicitly correlated F12b method produced energies in close agreement with the estimated CBS limit. A second fit to 1757 data at the CCSD(T)-F12b/VTZ-F12 level was constructed with an estimated fitting error of less than 1.5 cm(-1). The second surface has a global nonpolar O-in minimum, two T-shaped N-in minima, and two polar minima. Barriers between these minima are small and some wave functions have amplitudes in several wells. Low-lying rovibrational wave functions and energy levels up to about 150 cm(-1) were computed on the F12b PES using a discrete variable representation/finite basis representation method. Calculated rotational constants and intermolecular frequencies are in very close agreement with experiment.

Weight Function Control of Moving Least-Squares Interpolants: Application to Axisymmetric Shielding Current Analysis in HTS

The modified basis functions of element-free Galerkin (EFG) method generated by weight function control of moving least-squares interpolants for an axisymmetric problem are investigated. In the axisymmetric problem, a physical quantity of rotating angle direction on the symmetry axis should be zero. By using standard EFG, however, the values on symmetric axis are non zero because the axisymmetric is not considered in basis function. In order to clear up the above problem, the modified basis function of EFG is proposed. The numerical simulation of shielding current density in axisymmetric high-temperature superconductor is adopted for practical applications of modified axisymmetric EFG. The results of computations show that the shielding current density on symmetric axis vanish in both cases. In addition, the experimental critical current density measurement method is reproduced numerically by using the numerical code.

Improved Element-Free Galerkin method (IEFG) for solving three-dimensional elasticity problems

The essential idea of the element-free Galerkin method (EFG) is that moving least-squares (MLS) approximation are used for the trial and test functions with the variational principle (weak form). By using the weighted orthogonal basis function to construct the MLS interpolants, we derive the formulae for an improved element-free Galerkin (IEFG) method for solving three-dimensional problems in linear elasticity. There are fewer coefficients in improved moving least-squares (IMLS) approximation than in MLS approximation. Also fewer nodes are selected in the entire domain with the IEFG method than is the case with the conventional EFG method. In this paper, we selected a few example problems to demonstrate the applicability of the method.

An interpolating boundary element-free method (IBEFM) for elasticity problems

The paper begins by discussing the interpolating moving least-squares (IMLS) method. Then the formulae of the IMLS method obtained by Lancaster are revised. On the basis of the boundary element-free method (BEFM), combining the boundary integral equation method with the IMLS method improved in this paper, the interpolating boundary element-free method (IBEFM) for two-dimensional elasticity problems is presented, and the corresponding formulae of the IBEFM for two-dimensional elasticity problems are obtained. In the IMLS method in this paper, the shape function satisfies the property of Kronecker δ function, and then in the IBEFM the boundary conditions can be applied directly and easily. The IBEFM is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution to the nodal variables. Thus it gives a greater computational precision. Numerical examples are presented to demonstrate the method.

An improved boundary element-free method (IBEFM) for two-dimensional potential problems

The interpolating moving least-squares (IMLS) method is discussed first in this paper. And the formulae of the IMLS method obtained by Lancaster are revised. Then on the basis of the boundary element-free method (BEFM), combining the boundary integral equation (BIE) method with the IMLS method, the improved boundary element-free method (IBEFM) for two-dimensional potential problems is presented, and the corresponding formulae of the IBEFM are obtained. In the BEFM, boundary conditions are applied directly, but the shape function in the MLS does not satisfy the property of the Kronecker δ function. This is a problem of the BEFM, and must be solved theoretically. In the IMLS method, when the shape function satisfies the property of the Kronecker δ function, then the boundary conditions, in the meshless method based on the IMLS method, can be applied directly. Then the IBEFM, based on the IMLS method, is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied directly and easily, thus it gives a greater computational precision. Some numerical examples are presented to demonstrate the method.

Periodic boundary conditions in element free Galerkin method

PurposeThe purpose of this paper is to introduce a new methodology to implement periodic and anti‐periodic boundary conditions in the element free Galerkin method (EFGM).Design/methodology/approachThis paper makes use of the interpolating moving least squares (IMLS) in the EFGM to implement periodic and anti‐periodic boundary conditions. This fact allows imposing periodic and anti‐periodic boundary conditions in a way similar to the one used by the finite element method.FindingsEFGM generally uses the moving least squares to obtain its shape functions. So, these functions do not possess the Kronecker delta property. As a consequence, the imposition of essential, as well as periodic and anti‐periodic boundary conditions needs other techniques to do it. When EFGM makes use of IMLS the shape functions satisfy the Kronecker delta property. As consequence the periodic boundary conditions implementation can be done in a direct way, similar to the FEM.Originality/valueIMLS provides a new way of periodic boundary conditions implementation in EFGM. This kind of implementation provides an easy and direct way in comparison to usual existing methods. With this technique EFGM can now easily take advantage of electrical machines symmetry.

Interpolating moving least-squares methods for fitting potential energy surfaces: Using classical trajectories to explore configuration space

We develop two approaches for growing a fitted potential energy surface (PES) by the interpolating moving least-squares (IMLS) technique using classical trajectories. We illustrate both approaches by calculating nitrous acid (HONO) cis-->trans isomerization trajectories under the control of ab initio forces from low-level HF/cc-pVDZ electronic structure calculations. In this illustrative example, as few as 300 ab initio energy/gradient calculations are required to converge the isomerization rate constant at a fixed energy to approximately 10%. Neither approach requires any preliminary electronic structure calculations or initial approximate representation of the PES (beyond information required for trajectory initial conditions). Hessians are not required. Both approaches rely on the fitting error estimation properties of IMLS fits. The first approach, called IMLS-accelerated direct dynamics, propagates individual trajectories directly with no preliminary exploratory trajectories. The PES is grown "on the fly" with the computation of new ab initio data only when a fitting error estimate exceeds a prescribed tight tolerance. The second approach, called dynamics-driven IMLS fitting, uses relatively inexpensive exploratory trajectories to both determine and fit the dynamically accessible configuration space. Once exploratory trajectories no longer find configurations with fitting error estimates higher than the designated accuracy, the IMLS fit is considered to be complete and usable in classical trajectory calculations or other applications.

Application of Interpolating Moving Least Squares Fitting to Hypervelocity Collision Dynamics: O(3P) + HCl

We use an automated interpolating moving least-squares (IMLS) algorithm, which generates a fitted ab initio surface for systems of arbitrary topology, to construct a global OHCl ((3)A'') surface at the UB3LYP/aug-cc-pVTZ level of theory. This analytic PES includes all reaction channels and OHCl geometries with energies up to 144 kcal/mol (6.25 eV) above the O + HCl asymptote. The fitted surface was combined with the quasiclassical trajectory method to study the dynamics of the O((3)P) + HCl reaction at hyperthermal collision energies. The fitted PES greatly improves energy conservation during trajectory integration and eliminates problems with ab initio convergence, which are often encountered during direct dynamics studies. The more extensive trajectory calculations yield new insight into the title reaction and agree well with previous experimental studies and direct dynamics results.

Haptic quantum chemistry

We present an implementation designed to physically experience quantum mechanical forces between reactants in chemical reactions. This allows one to screen the profile of potential energy surfaces for the study of reaction mechanisms. For this, we have developed a interface between the user and a virtual laboratory by means of a force-feedback haptic device. Potential energy surfaces of chemical reactions can be explored efficiently by rendering in the haptic device the gradients calculated with first-principles methods. The underlying potential energy surface is accurately fitted on the fly by the interpolating moving least-squares (IMLS) scheme to a grid of quantum chemical electronic energies (and geometric gradients). In addition, we introduce a new IMLS-based method to locate minimum-energy paths between two points on a potential energy surface.

Collocation Discrete Least Square (CDLS) Method for Elasticity Problems and Grid Irregularity Effect Assessment

A meshless approach, collocation discrete least square (CDLS) method, is extended in this paper, for solving elasticity problems and grid irregularity effect is assessed. In the present CDLS method, the problem domain is discretized by distributed field nodes. The field nodes are used to construct the trial functions. The moving least-squares interpolant is employed to construct the trial functions. Some collocation points that can be independent of the field nodes are used to form the total residuals of the problem. The least-squares technique is used to obtain the solution of the problem by minimizing the summation of the residuals for the collocation points. The final stiffness matrix is symmetric and therefore can be solved directly via efficient solvers. The boundary conditions are easily enforced by the penalty method. The present method does not require any mesh so it is a truly meshless method. Numerical examples are studied in detail, which show that the present method is stable and possesses good accuracy, high convergence rate and high efficiency for both regular and irregular point distribution.

Improved element-free Galerkin method for two-dimensional potential problems

Potential difficulties arise in connection with various physical and engineering problems in which the functions satisfy a given partial differential equation and particular boundary conditions. These problems are independent of time and involve only space coordinates, as in Poisson's equation or the Laplace equation with Dirichlet, Neuman, or mixed conditions. When the problems are too complex, they usually cannot be solved with analytical solutions. The element-free Galerkin (EFG) method is a meshless method for solving partial differential equations on which the trial and test functions employed in the discretization process result from moving least-squares (MLS) interpolants. In this paper, by using the weighted orthogonal basis function to construct the MLS interpolants, we derive the formulae of an improved EFG (IEFG) method for two-dimensional potential problems. There are fewer coefficients in the improved MLS (IMLS) approximation than in the MLS approximation, and in the IEFG method fewer nodes are selected in the entire domain than in the conventional EFG method. Hence, the IEFG method should result in a higher computing speed.

Meshless numerical method based on tensor product

A normalized space constructed by tensor product is used in field function approach to give a special case of moving least squares (MLS) interpolation scheme. In the regular domain, the field function which meets homogenous boundary conditions is constructed by spanning base space to make the MLS interpolation scheme simpler and more efficient. Owing to expanded basis functions selection, some drawbacks in general MLS method, for example repeated inversion, low calculation efficiency, and complex criterions, can be avoided completely. Numerical examples illustrate that the proposed method is characterized by simple mathematical concept, convenient repeat calculations with high accuracy, good continuity, less computation and rapid convergence.