A grid based particle method for moving interface problems

We propose a novel algorithm for modeling interface motions. The interface is represented and is tracked using quasi-uniform meshless particles. These particles are sampled according to an underlying grid such that each particle is associated to a grid point which is in the neighborhood of the interface. The underlying grid provides an Eulerian reference and local sampling rate for particles on the interface. It also renders neighborhood information among the meshless particles for local reconstruction of the interface. The resulting algorithm, which is based on Lagrangian tracking using meshless particles with Eulerian reference grid, can naturally handle/control topological changes. Moreover, adaptive sampling of the interface can be achieved easily through local grid refinement with simple quad/oct-tree data structure. Extensive numerical examples are presented to demonstrate the capability of our new algorithm.     

The adaptive immersed interface finite element method for elliptic and Maxwell interface problems

We propose self-adaptive finite element methods with error control for solving elliptic and electromagnetic problems with discontinuous coefficients. The meshes in the methods do not need to fit the interfaces. New error indicators are introduced to control the error due to non-body-fitted meshes. Flexible h-adaptive strategies are developed, which can be systematically extended to a large class of interface problems. Extensive numerical experiments are performed to support the theoretical results and to show the competitive behavior of the adaptive algorithm even for interfaces involving corner or tip singularities.     

Inverse scattering for geophysical problems

A method for finding wave velocity from the scattering data is given for three-dimensional problems.     

Augmented coupling interface method for solving eigenvalue problems with sign-changed coefficients

In this paper, we propose an augmented coupling interface method on a Cartesian grid for solving eigenvalue problems with sign-changed coefficients. The underlying idea of the method is the correct local construction near the interface which incorporates the jump conditions. The method, which is very easy to implement, is based on finite difference discretization. The main ingredients of the proposed method comprise (i) an adaptive-order strategy of using interpolating polynomials of different orders on different sides of interfaces, which avoids the singularity of the local linear system and enables us to handle complex interfaces; (ii) when the interface condition involves the eigenvalue, the original problem is reduced to a quadratic eigenvalue problem by introducing an auxiliary variable and an interfacial operator on the interface; (iii) the auxiliary variable is discretized uniformly on the interface, the rest of variables are discretized on an underlying rectangular grid, and a proper interpolation between these two grids are designed to reduce the number of stencil points. Several examples are tested to show the robustness and accuracy of the schemes.     

Inverse problems for matrix Sturm-Liouville operators

Inverse spectral problems for nonselfadjoint matrix Sturm-Liouville differential operators on a finite interval and on the half-line are studied. As a main spectral characteristic, we introduce the so-called Weyl matrix and prove that the specification of the Weyl matrix uniquely determines the matrix potential and the coefficients of the boundary conditions. Moreover, for a finite interval, we also study the inverse problems of recovering matrix Sturm-Liouville operators from discrete spectral data (eigenvalues and “weight” numbers) and from a system of spectra. The results thus obtained are natural generalizations of the classical results in inverse problem theory for scalar Sturm-Liouville operators. Dedicated to the memory of B. M. Levitan     

Inverse problems of aggregation processes

The coagulation frequency is the key ingredient in the population balance (Smoluchowski) equation of coagulation kinetics. An inverse problem is formulated to extract the coagulation frequency from transient size distributions when these distributions are self-similar. Two numerical examples illustrate the procedure. The first demonstrates the inverse problem for the recovery of singular coagulation frequencies, while the second shows the procedure when self-similarity is approximate. Transient droplet coagulation experiments in a turbulent flow field have been performed. The resulting size distributions are observed to be self-similar. The inverse problem is used to determine the drop coagulation frequency. This frequency shows significant deviation from the coagulation frequencies derived from simple models of drop-drop interactions in a turbulent flow field.     

Three-dimensional elliptic solvers for interface problems and applications

Second-order accurate elliptic solvers using Cartesian grids are presented for three-dimensional interface problems in which the coefficients, the source term, the solution and its normal flux may be discontinuous across an interface. One of our methods is designed for general interface problems with variable but discontinuous coefficient. The scheme preserves the discrete maximum principle using constrained optimization techniques. An algebraic multigrid solver is applied to solve the discrete system. The second method is designed for interface problems with piecewise constant coefficient. The method is based on the fast immersed interface method and a fast 3D Poisson solver. The second method has been modified to solve Helmholtz/Poisson equations on irregular domains. An application of our method to an inverse interface problem of shape identification is also presented. In this application, the level set method is applied to find the unknown surface iteratively.     

Inverse substructure method for model updating of structures

Traditional model updating of large-scale structures is usually time-consuming because the global structural model needs to be repeatedly re-analyzed as a whole to match global measurements. This paper proposes a new substructural model updating method. The modal data measured on the global structure are disassembled to obtain the independent substructural dynamic flexibility matrices under force and displacement compatibility conditions. The method is extended to the case when the measurement is carried out at partial degrees-of-freedom of the structure. The extracted substructural flexibility matrices are then used as references for updating the corresponding substructural models. An orthogonal projector is employed on both the extracted substructural measurements and the substructural models to remove the rigid body modes of the free–free substructures. Compared with the traditional model updating at the global structure level, only the sub-models at the substructural level are re-analyzed in the proposed substructure-based model updating process, resulting in a rapid convergence of optimization. Moreover, only measurement on the local area corresponding to the concerned substructures is required, and those on other components can be avoided. The effectiveness and efficiency of the proposed substructuring method are verified through applications to a laboratory-tested frame structure and a large-scale 600 m tall Guangzhou New TV Tower. The present technique is referred to as the inverse substructuring model updating method as the measured global modal data are disassembled into the substructure level and then the updating is conducted on the substructures only. This differs from the substructuring model updating method previously proposed by the authors, in which the model updating is still conducted in the global level and the numerical global modal data are assembled from those of substructures. That can be referred to as the forward substructuring model updating method.     

Inverse problems in mass calculations of atomic parameters

Methods of solving inverse problems which arise in determining atomic parameters are briefly reviewed. The parametric method of studying serial perturbations in the spectra of complex elements is considered in more detail. The classification of interacting series complexes given here, together with the generalized Edlen serial formula and the corresponding results, modifies existing concepts regarding the role of interactions of different series.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 25–30, April, 1991.     

A Correction Function Method for Poisson problems with interface jump conditions

In this paper we present a method to treat interface jump conditions for constant coefficients Poisson problems that allows the use of standard “black box” solvers, without compromising accuracy. The basic idea of the new approach is similar to the Ghost Fluid Method (GFM). The GFM relies on corrections applied on nodes located across the interface for discretization stencils that straddle the interface. If the corrections are solution-independent, they can be moved to the right-hand-side (RHS) of the equations, producing a problem with the same linear system as if there were no jumps, only with a different RHS. However, achieving high accuracy is very hard (if not impossible) with the “standard” approaches used to compute the GFM correction terms.     

Inverse problem method in laser physics

Applications of the method of ill-posed inverse problems in laser physics are considered. Means of calculating the level cross sections of physical-chemical reactions with subsequent calculation of the reaction rate constants are developed. Mathematical and numerical aspects for the realization of ill-posed inverse problem are considered in detail using the concrete example of level cross sections of dissociation.     

Resonating-group method for nuclear many-body problems

The resonating-group method is a microscopic method which uses fully antisymmetric wave functions, treats correctly the motion of the total center of mass, and takes cluster correlations into consideration. In this review, we discuss the formulation of this method for various nuclear many-body problems, and describe a complex-generator-coordinate technique which has been employed to evaluate matrix elements required in resonating-group calculations. Several illustrative examples of bound-state, scattering, and reaction calculations, which serve to demonstrate the usefulness of this method, are presented. Finally, by utilizing the results of these calculations, we discuss the role played by the Pauli principle in nuclear scattering and reaction processes.     

The imaginary-time method for relativistic problems

A relativistic version of the imaginary-time method is presented. The method is used to calculate the probability w of ionization of a bound state by electric and magnetic fields of various configurations (including the case when the binding energy E _b is comparable to mc ₂). The formulas cover as limiting cases both the ionization of nonrelativistic bound systems (atoms and ions) and the case E _b =2mc ₂, when w equals the probability of electron-positron pair production from the vacuum in the presence of a strong field. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 4, 213–218 (25 August 1997)     

Anti-diffusion method for interface steepening in two-phase incompressible flow

In this paper, we present a method for obtaining sharp interfaces in two-phase incompressible flows by an anti-diffusion correction, that is applicable in a straight-forward fashion for the improvement of two-phase flow solution schemes typically employed in practical applications. The underlying discretization is based on the volume-of-fluid (VOF) interface-capturing method on unstructured meshes. The key idea is to steepen the interface, independently of the underlying volume-fraction transport equation, by solving a diffusion equation with reverse time, i.e. an anti-diffusion equation, after each advection time step of the volume fraction. As the solution of the anti-diffusion equation requires regularization, a limiter based on the directional derivative is developed for calculating the gradient of the volume fraction. This limiter ensures the boundedness of the volume fraction. In order to control the amount of anti-diffusion introduced by the correction algorithm we propose a suitable stopping criterion for interface steepening. The formulation of the limiter and the algorithm for solving the anti-diffusion equation are applicable to 3-dimensional unstructured meshes. Validation computations are performed for passive advection of an interface, for 2-dimensional and 3-dimensional rising-bubbles, and for a rising drop in a periodically constricted channel. The results demonstrate that sharp interfaces can be recovered reliably. They show that the accuracy is similar to or even better than that of level-set methods using comparable discretizations for the flow and the level-set evolution. Also, we observe a good agreement with experimental results for the rising drop where proper interface evolution requires accurate mass conservation.     

Gradient-augmented hybrid interface capturing method for incompressible two-phase flow

Motivated by inconveniences of present hybrid methods,a gradient-augmented hybrid interface capturing method(GAHM) is presented for incompressible two-phase flow.A front tracking method(FTM) is used as the skeleton of the GAHM for low mass loss and resources.Smooth eulerian level set values are calculated from the FTM interface,and are used for a local interface reconstruction.The reconstruction avoids marker particle redistribution and enables an automatic treatment of interfacial topology change.The cubic Hermit interpolation is employed in all steps of the GAHM to capture subgrid structures within a single spacial cell.The performance of the GAHM is carefully evaluated in a benchmark test.Results show significant improvements of mass loss,clear subgrid structures,highly accurate derivatives(normals and curvatures) and low cost.The GAHM is further coupled with an incompressible multiphase flow solver,Super CE/SE,for more complex and practical applications.The updated solver is evaluated through comparison with an early droplet research.     

Optimum and standard beam widths for numerical modeling of interface scattering problems

Gaussian beams provide a useful insonifying field for surface or interface scattering problems such as encountered in electromagnetics, acoustics and seismology. Gaussian beams have these advantages: (i) They give a finite size for the scattering region on the interface. (ii) The incident energy is restricted to a small range of grazing angles. (iii) They do not have side lobes. (iv) They have a convenient mathematical expression. The major disadvantages are: (i) Insonification of an interface is nonuniform. The scattered field will depend on the location of the scatterers within the beam. (ii) The beams spread, so that propagation becomes an integral component of the scattering problem. A standard beam parameterization is proposed which keeps propagation effects uniform among various models so that the effects of scattering only can be compared. In continuous wave problems, for a given angle of incidence and incident amplitude threshold, there will be an optimum Gaussian beam which keeps the insonified area as small as possible. For numerical solutions of pulse beams, these standard parameters provide an estimate of the smallest truncated domain necessary for a physically meaningful result.     

An Inverse method of elastic constants for unidirectional fiber-reinforced composite plate

An inverse method is presented to determine the elastic constants of an experimental sample, a titanium graphite unidirectional fiber-reinforced composite plate, using wavelet transform and neural networks. Optimal algorithms of wavelet transform and neural networks are given here in order to improve the accuracy of inversion results. Coherent results were shown in both fiber direction and cross fiber direction, proving the feasibility of this method. Neither the group velocity of the Lamb wave modes are needed, as in the conventional method, and no direct least-square fitting of the experimental waveforms is necessary. __________ Translated from Acta Acustica, 2005, 30(4) (in Chinese)     

Inverse problem method for the perturbed nonlinear Schrödinger equation

A scattering matrix is defined for eq. (1), and a complete set of equations, governing the time evolution of the scattering parameters, is derived by a general method. These equations form a basis for the perturbation theory.