Quantum scattering using a novel implementation based on the variational R matrix formalism and the finite element method: a comparative study

The functionals of the Kohn and Sil variational principles of atomic collision theory contain second-order spatial derivatives of the wave-function. Ipso facto the "second-order" generalised Euler-Lagrange differential equation is a necessary condition for these functionals to yield a stationary value. To illustrate this we provide a classical prescription for the time-dependent and time-independent quantum potential scattering problems, generalising the latter to the three-body problem. As practical examples where the presence of second-order derivatives in the Lagrangian density makes a non-trivial contribution to the equations of motion, we consider the optimisation of translation factors in a trial wave-function using the Sil variational principle and the optimisation of a variable charge using the Kohn variational principle. As further examples of Lagrangian densities where second-order derivatives may occur we briefly mention electromagnetism and relativistic quantum mechanics.

ODPEVP: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm–Liouville problem

The handling and control of towed cable and body systems onboard surface ships and submarines presents a significant technical challenge to design engineers in the defense and ocean industries. The current approaches rely heavily on the empirical methods and the time-consuming and costly prototype testing. Computer simulation provides a cost effective way to reduce the high risks associated with the towed cable/body system. However, the current dynamic analysis of towed cables is mostly done by the finite difference (FD) method in stead of the finite element (FE) method that is widely used in almost all engineering fields. This paper presents an alternative FE method to simulate the dynamics of towed cable and body system, in which the large rigid body motion is coupled with small elastic deformation. The newly derived FE method is formulated in terms of element nodal positions, which is different from the existing FE methods that use displacements. The alternative FE method solves for the cable position directly in order to eliminate accumulated numerical errors arising from existing FE methods that solve for displacements first in order to obtain the cable position over very long period of time. The alternative FE formulation is implemented and applied to real applications to demonstrate its robustness by comparing simulation results with the experimental and sea trial data.

We propose new basis sets for linear algebraic variational calculations of transition amplitudes for quantum-mechanical scattering problems. These basis sets are hybrids of those that yield the Kohn variational principle (KVP) and those that yield the generalized Newton variational principle (GNVP) when substituted in Schlessinger's stationary expression for the T operator. Trial calculations show that efficiencies almost as great as that of the GNVP and much greater than the KVP can be obtained even for basis sets with the majority of the members independent of energy.

Real-world groundwater modeling deals with heterogeneous and anisotropic aquifer systems. Numerical models, particularly the finite difference and finite element methods, are extensively used for modeling such a complex aquifer system. The finite difference method is an age-old straightforward approach, whereas the finite element method is more idealistic, relatively new, and has emerged as a popular approach. The unique advantages with the finite element method are (i) an irregular-shaped domain can be discretized into elements with a matching boundary, (ii) a node can be connected with the neighboring nodes in any direction. Thus, the finite element analysis allows water/solute to flow more freely toward the surrounding nodes along nodal links. On the other hand, the mathematical formulation of the finite element technique follows a set of rules step by step. Conventionally, this idealistic approach uses shape functions and integration techniques to develop element matrices, which, in turn, are assembled to form the global matrix. This conventional global matrix was found to be inconsistent with the basic partial differential equation representing the mass balance in a nodal domain as a control volume. As a result, the matrix solution, particularly for nonlinear problems (e.g., vadose zone flow, density dependent flow, and so forth) becomes unstable and computationally burdensome. To overcome the limitations of the conventional finite element method for real-world groundwater modeling, a new approach, called the direct-formulation finite element (DFFE) method, has been developed. In this approach, a two-dimensional model domain is discretized into triangular elements. Element matrices are formed directly based on flow principles and material properties of elements without using shape functions and integration process. The DFFE method is robust. The global matrix is consistent, and the solution is stable and efficient. The results obtained from this approach were found in a good agreement with that obtained from analytical and other widely used models.

In this article, I consider the use of the Kohn variational principle in obtaining the reaction matrix, which determines the positive energy solution to the partial wave Schrodinger equation in coordinate space. The formulation considered is appropriate for electron scattering from an atom or molecule. In its usual form, the Kohn variational principle provides the relationship between the exact reaction matrix, independent of first-order errors in the trial wavefunction, and the trial reaction matrix. This relationship is usually called the Kato identity. It is obtained using the second form of Green's theorem for a sphere of infinite radius. I show how a Kohn-like variational principle, using Green's theorem for a sphere of arbitrary radius, provides an expression for the exact, coordinate space, integral equation amplitudes in terms of the trial amplitudes. The results are again exact to second order in the trial wavefunction. Applications of the present results to electron-molecule frame transformations and algebraic scattering theory are then suggested.

It is proved that if a scaling parameter alpha contained in suitably chosen trial functions F1T and F2T is optimized using Kohn's variational principle, then the virial theorem is satisfied by the optimized variational functions and the variationally determined scattering amplitude. A generalization of the theorem to arbitrary values of the scaling parameter is also established. In addition the virial theorem for the scattering amplitude is applied to the elastic scattering of electrons by atomic hydrogen.

Objective To explore the establishment of 3D thoracic model by finite element methods, and study the mechanical mechanism of minimally invasive surgery for correcting the chest wall deformity, and provide personalized surgery solution in the future. Methods According to admission and exclusion criteria, we selected 10 cases of pectus carinatum that received chest CT scan. The finite element model of pectus carinatum was established and analyzed by Mimics, ABAQUS, etc. The validity of finite element method for chest wall was verified by comparing the sternal displacement between the simulated values and actual values with the same force. Results The 3D finite element model of pectus carinatum was successfully established and analyzed. The stress distribution of the 10 cases in the posterior ribs was mainly in the 1-6 ribs on both sides, mostly concentrated in the 4th to 6th ribs, and the stress was symmetrical on both sides. Statistical analysis showed that the displacement value of the sternum was correlated, and the validity of the model was verified. Conclusion Mimics, ABAQUS and other finite element modeling and analysis software can effectively establish the thoracic 3D finite element model and mechanical analysis, which can help the further development of personalized minimally invasive surgery for correcting chest wall deformities. Key words: Pectus carinatum; Chest wall deformity; Finite element analysis

The scientific basis of orthodontics is physics and Newtonian mechanics applied to the tooth-periodontal ligament-bone biologic system. The mechanical response of the periodontal ligament (PDL) to forces has a significant impact on bone remodelling and tooth movement. At the early stages, orthodontists mainly evaluated the PDL mechanical properties by in vitro experiments, which guided the use of forces in the orthodontic process. With the development of computers, finite element (FE) methods were increasingly used in the study of orthodontics. The phenomena and results of PDL mechanical tests are abstracted into constitutive equations. The identified constitutive equations are then implemented in finite element software. The finite element method could be used to simulate in vitro and in vivo experiments and predict the response of PDL under various complex working conditions. A visco-hyperelastic constitutive model is proposed and implemented in finite element technique based on prior static mechanical experiments of human PDL. Some complex experimental conditions are simulated to discuss the properties of the constitutive model.

It has been well known that exact closed-form solutions are not available for non-Levy-type plates. Thus, more accurate and efficient computational methods have been required for the plates subjected to arbitrary boundary conditions. This paper presents a frequency-domain spectral element model for the rectangular finite plate element. The spectral element model is developed by using two methods in combination: (1) the boundary splitting and (2) the super spectral element method in which the Kantorovich method-based finite strip element method and the frequency-domain waveguide method are utilized. The present spectral element model has nodes on four edges of the finite plate element, but no nodes inside. This can reduce the total number of degrees of freedom a lot to improve the computational efficiency significantly, when compared with the standard finite element method (FEM). The high solution accuracy and computational efficiency of the present spectral element model are evaluated by the comparison with exact solutions and the solutions by the standard FEM.

Finite element method(FEM) is one of the most popularly used method analyzing the dynamic behaviors of structures. But unless number of finite elements is large enough, the results from FEM some what different from exact analytical solutions, especially at high frequency range. On the other hand, as the spectral analysis method(SAM) deals directly with the governing equations of a structure, the results from this melthod cannot but be exact regardless of any frequency range. However, the SAM can be applied only to the case where a structure is subjected to the concentrated loads, despite a structure could be unddergone distributed loads more generally. In this paper, therefore, new spectral analysis algorithm is introduced through the spectral element method(SEM), so that it can be applied to anlystructures whether they are subjected to the concentrated loads or to the distributed loads. The results from this new SEM are compared with both the results from FEM and the exact analytical solutions. As expected, the results from new SEM algorithm are found to be almost identical to the exact analytical solutions while those from FEM are not agreed well with the exact analytical solutions as the mode number increases.

Objectives: This paper proposes a general formula for computation of the stiffness matrix of three-dimensional linearelastic bodies by the method of Moment Schema of Finite Elements (MSFE). Methods: Numerical methods based on Lagrange variational principle, such as Finite Element Method (FEM) and the method of Moment Schema of Finite Elements (MSFE). The paper focuses on the problem of obtaining the stiffness matrices of Finite Elements (FE), which ensure effectiveness of the developed version of FEM. Variational principle for calculation of energy functional of rectangular three-dimensional finite element is used. Results: Analyses of various problems of mechanics of deformable solids shows slow convergence of traditional FEM, especially for rigid bodies and surfaces of complex curved shapes. This paper proposes a scheme for inference for FEM ratios, which takes into account the basic properties of rigid displacements for isoparametric and curvilinear FE. The presented version of FEM allows us to obtain the stiffness matrix of FE, taking into account the effect of a false shift and low compressibility of elastomers. Conclusion/Application: The general method to find the potential energy of three-dimensional linear-elastic bodies is proposed, which allows us to solve the problems of mechanics of complex structures with higher accuracy.

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

The performance of electrical machines is largely dictated by the action of current and flux in the core length. The field in a cross-section obeys Poisson's equation and approximate solutions have been obtained by finite difference and element methods. The finite difference method requires a large number of nodes and is slow to converge as permeability is variable. The finite element method is more flexible being more readily fitted to iron-air boundaries and has better convergence. However, it is difficult to formulate a legitimate variational formulation for transient conditions in the presence of dissipation. Here, discrete equations are formed by applying Ampere's circuital law around each node. Careful choice of contour lines give a current distribution superior to that obtained with finite elements. Fast convergence is obtained and the method is applicable under transient conditions.

We consider three types of subdiffusion models, namely single-term, multi-term and distributed order fractional diffusion equations, for which the maximum-principle holds and which, in particular, preserve nonnegativity. Hence the solution is nonnegative for nonnegative initial data. Following earlier work on the heat equation, our purpose is to study whether this property is inherited by certain spatially semidiscrete and fully discrete piecewise linear finite element methods, including the standard Galerkin method, the lumped mass method and the finite volume element method. It is shown that, as for the heat equation, when the mass matrix is nondiagonal, nonnegativity is not preserved for small time or time-step, but may reappear after a positivity threshold. For the lumped mass method nonnegativity is preserved if and only if the triangulation in the finite element space is of Delaunay type. Numerical experiments illustrate and complement the theoretical results.