Eigen Solution Research Articles

Stiffness tensor estimation of anisotropic crystal using point contact method and unscented Kalman filter

The potential application of Lithium Niobate (LiNbO3) crystal is immense, specifically in the domain of meta-surfaces and nano-resonators. However, the practical application of LiNbO3 is impeded due to unreliable experimental techniques and inaccurate inversion algorithms for material characterization. In the current research, material characterization of anisotropic crystal is proposed by exploring the wavefield evolution in the spatial and temporal domains. The presented framework has three major components: a physics-based mathematical model (Christoffel equation), a novel experimental technique, and an inversion algorithm based on Bayesian filtering. An experimental technique based on Coulomb coupling is devised to visualize the propagation of ultrasonic waves in an anisotropic crystal. The crystal is characterized by measuring the directional-dependent acoustic wave velocity from the spatial–temporal information of the wave propagation. The anisotropic constitutive properties of the crystal are estimated by exploring the wave velocity in the Bayesian filtering algorithm. The proposed algorithm is based on the probabilistic framework that integrates the experimental measurement in a physics-based mathematical model for optimal state prediction of stiffness tensor through the Bayesian filtering algorithm. In particular, we utilize the unscented Kalman filter (UKF) in conjunction with the plane-wave Eigen solution to estimate the constitutive parameters. In the presence of measurement uncertainties, the performance of the optimal prediction algorithm is illustrated by comparing the estimated parameter with the corresponding theoretical value. The comparison demonstrates that the proposed inversion algorithm is efficient and robust and performs satisfactorily even with significant measurement uncertainties.

A novel super symplectic analytical singular element for crack propagation along a bimaterial interface

In the last few years, the corresponding author and colleagues proposed a type of symplectic analytical singular elements (SASEs) for various crack problems, such as cracks in plate and shell, bimaterial crack, dynamic crack, crack in viscoelastic media, etc. However, the SASE is still confined to stationary cracks. In this contribution, we propose a new super symplectic analytical singular element (SSASE) for crack propagation along a bimaterial interface. The shape of the SSASE is extended from the normally used circle to general convex polygon, and it can work well on a fixed mesh with triangles or quadrilaterals, thus more proper for modelling crack propagation. The proposed element uses the symplectic eigen solution to define its internal fields, thus most of the advantages of the SASE family are reserved. The proposed element is verified and validated through a few numerical examples. The present contribution has established a new platform for the modelling of the progressive failure process along a bimaterial interface and has also extended the application of the analytical symplectic dual approach to the numerical approach for crack propagations.

Electron wave spin in a quantum well

The particle-wave duality of the electron poses a principle question of whether the spin is a property of the particle or the wave. In this paper, the wave nature of the spin is studied for an electron inside a two-dimensional quantum well. By solving the exact 4 − spinor eigen solution to the Dirac equation, we show that a stable circulating total current density exists inside the well with a donut shaped topography. A spin value is modified by the confining geometry of the well. Our analysis also shows that a free electron Gaussian wavepacket is unstable and experiences quick decoherence.

On the symplectic superposition method for free vibration of rectangular thin plates with mixed boundary constraints on an edge

A novel symplectic superposition method has been proposed and developed for plate and shell problems in recent years. The method has yielded many new analytic solutions due to its rigorousness. In this study, the first endeavor is made to further developed the symplectic superposition method for the free vibration of rectangular thin plates with mixed boundary constraints on an edge. The Hamiltonian system-based governing equation is first introduced such that the mathematical techniques in the symplectic space are applied. The solution procedure incorporates separation of variables, symplectic eigen solution and superposition. The analytic solution of an original problem is finally obtained by a set of equations via the equivalence to the superposition of some elaborated subproblems. The natural frequency and mode shape results for representative plates with both clamped and simply supported boundary constraints imposed on the same edge are reported for benchmark use. The present method can be extended to more challenging problems that cannot be solved by conventional analytic methods.

An oblique circular cylinder element for 3D interfacial cracks in composites

This study further extends a developed symplectic analytical singular element (SASE) library to include three-dimensional (3D) cracks in dissimilar materials. We construct a trial symplectic eigen solution for 3D bimaterial interfacial cracks and use it as the crack tip enrichment to define the interior displacement field of a new singular element. The symplectic approach is applied for the numerical modelling of 3D bimaterial cracks for the first time. In the derivation, a non-standard shape of the 3D crack-tip element, i.e., a cylinder with two oblique ends with arbitrary orientations, is considered to better fit the mesh of curved 3D cracks. It is proven that the proposed 3D element has improved the accuracy and is still convenient to use. The present contribution has provided a new insight for choosing effective and efficient crack tip enrichments of 3D interfacial cracks, and is also an important step forward of the SASE library.

Solar large-scale magnetic field and cycle patterns in solar dynamo

ABSTRACT We compare spectra of the zonal harmonics of the large-scale magnetic field of the Sun using observation results and solar dynamo models. The main solar activity cycle as recorded in these tracers is a much more complicated phenomenon than the eigen solution of solar dynamo equations with the growth saturated by a back reaction of the dynamo-driven magnetic field on solar hydrodynamics. The nominal 11(22)-yr cycle as recorded in each mode has a specific phase shift varying from cycle to cycle; the actual length of the cycle varies from one cycle to another and from tracer to tracer. Both the observation and the dynamo model show an exceptional role of the axisymmetric ℓ5 mode. Its origin seems to be readily connected with the formation and evolution of sunspots on the solar surface. The results of observations and dynamo models show a good agreement for the low ℓ1 and ℓ3 modes. The results for these modes do not differ significantly for the axisymmetric and non-axisymmetric models. Our findings support the idea that the sources of the solar dynamo arise as a result of both the distributed dynamo processes in the bulk of the convection zone and the surface magnetic activity.

Time domain flutter analysis of bend-twist coupled large composite wind turbine blades: a parametric study

This article presents flutter prediction of composite wind turbine blades through time domain aeroelastic analysis using unsteady blade element momentum (BEM) theory based aerodynamic model. Comparison of the flutter speeds obtained in the present study with those previously obtained through Eigen solution based approach using Theodorsen’s theory based aerodynamic model for several different lamination sequences and blade configuration has been presented. It has been observed that both the approaches predict highest flutter speed at the same blade configuration and lamination sequence. However, flutter limit obtained through the present approach is less as compared to that obtained through Eigen solution based approach.

A new crack-tip singular element for cracks in three-dimensional elastic bodies

Symplectic analytical singular element (SASE) is a special crack-tip element in the framework of finite element method (FEM) for modelling cracks. A group of SASEs, which have been developed by the authors, for various crack problems almost in two-dimensional (2D), but not touched yet more complex ones, i.e., in three-dimensional (3D) domain. The underlying reason lies in the fact that analytical symplectic eigen solution for 3D elastic crack problem is not yet available, a 3D SASE thus cannot be constructed through a simple generalization from the 2D SASEs. This study aims to fill out this 3D gap, we thus use a trail displacement field to construct a 3D SASE. The derived strain and stress fields still possess singularity in the vicinity of crack-tip. Finite element formulation is derived based on the minimum total potential energy principle. It is found that some of key features of a 2D SASE still exist in the developed 3D one, e.g., the SIFs can be calculated accurately without any post-processing. Numerical examples are considered to show the accuracy and performance of the proposed element.

Analytical Solution for Thermal Flutter of Laminates in Supersonic Speeds

As a basic component of engineering fields such as aeronautics, astronautics and shipbuilding, panel structure has been widely used in engineering and scientific research. It is of great theoretical and practical significance to study the vibration of panels. The panel flutter problem has caused widely concerned by researchers at home and abroad during to the emergence of high-speed aircrafts. With regard to the eigenvalue problem of rectangular panels, it is generally believed that it is difficult to obtain a closed form eigen solution in the case of an adjacent boundaries clamped-supported or a free boundary that cannot be decoupled. Aiming at the problem, this paper studies the two-dimensional symmetric orthogonal laminated plate structure in the hypersonic flow in the thermal environment, and combines the first-order piston aerodynamic theory to study a high-precision separation variable method. Through this method, analytical solution to the closed form of the thermal flutter problem of rectangular panels can be obtained under any homogeneous boundary conditions.

Eigen Solution of Neural Networks and Its Application in Prediction and Analysis of Controller Parameters of Grinding Robot in Complex Environments

The robot dynamic model is often rarely known due to various uncertainties such as parametric uncertainties or modeling errors existing in complex environments. It is a key problem to find the relationship between the changes of neural network structure and the changes of input and output environments and their mutual influences. Firstly, this paper defined the conceptions of neural network solution, neural network eigen solution, neural network complete solution, and neural network partial solution and the conceptions of input environments, output environments, and macrostructure of neural networks. Secondly, an eigen solution theory of general neural networks was proposed and proven including consistent approximation theorem, eigen solution existence theorem, consistency theorem of complete solution, the partial solution, and none solution theorem of neural networks. Lastly, to verify the eigen solution theory of neural networks, the proposed theory was applied to a novel prediction and analysis model of controller parameters of grinding robot in complex environments with deep neural networks and then build prediction model with deep learning neural networks for controller parameters of grinding robot. The morphological subfeature graph with multimoment was constructed to describe the block surface morphology using rugosity, standard deviation, skewness, and kurtosis. The results of theoretical analysis and experimental test show that the output traits have an optional effect with joint action. When the input features functioning in prediction increase, higher predicted accuracy can be obtained. And when the output traits involving in prediction increase, more output traits can be predicted. The proposed prediction and analysis model with deep neural networks can be used to find and predict the inherent laws of the data. Compared with the traditional prediction model, the proposed model can predict output features simultaneously and is more stable.

Some Characteristic Functions for the Eigen Solution of Nonlinear Spinor

In this paper, we solve the eigen solutions to some nonlinear spinor equations, and compute several functions reflecting their characteristics. The numerical results show that, the nonlinear spinor equation has only finite meaningful eigen solutions, which have positive discrete mass spectra and anomalous magnetic moment. The nonlinear potential and interactions yield different contributions to the total energy, and these components of the energy lead to different energy-speed relation. The magnitude of these components can be detected by elaborate experiments. The weird properties of the nonlinear spinors might be closely related with the elementary particles and their interactions, so some deeper investigations on them are significant.

A novel size independent symplectic analytical singular element for inclined crack terminating at bimaterial interface

• The analytical symplectic eigen expansion for inclined bimaterial interface crack is solved. • Stress intensity factors for the inclined bimaterial crack derived from the proposed method are provided and discussed. • A novel singular finite element is constructed based on the symplectic eigen solution. • It is proven that the stiffness matrix of the proposed singular element is independent of the element's size. • Stress intensity factors can be calculated directly without any post-processing. Cracks often exist in composite structures, especially at the interface of two different materials. These cracks can significantly affect the load bearing capacity of the structure and lead to premature failure of the structure. In this paper, a novel element for modeling the singular stress state around the inclined interface crack which terminates at the interface is developed. This new singular element is derived based on the explicit form of the high order eigen solution which is, for the first time, determined by using a symplectic approach. The developed singular element is then applied in finite element analysis and the stress intensity factors (SIFs) for a number of crack configurations are derived. It has been concluded that composites with complex geometric configurations of inclined interface cracks can be accurately simulated by the developed method, according to comparison of the results against benchmarks. It has been found that the stiffness matrix of the proposed singular element is independent of the element size and the SIFs of the crack can be solved directly without any post-processing.

A symplectic analytical singular element for the inclined crack terminating at the material interface of composite structures

The inclined bimaterial interface crack problem is studied from theoretical and computational aspects. The original problem is transformed into an eigen value problem by using the symplectic approach, and the eigen solution are obtained. A Symplectic Analytical Singular Element (SASE), of which the interior fields are described by the obtained eigen solutions, is developed. The stress intensity factors (SIFs) can be calculated directly without any post-processing. Numerical example on a square bimaterial plate containing an inclined crack is provided to validate the proposed method. The comparison of the predicted results with the existing result indicates that the proposed method is accurate for the modelling of the inclined bimaterial interface crack.

Study on steady-state thermal conduction with singularities in multi-material composites

Increasing demand in material and mechanical properties has led to production of complex composite structures. The composite structures, made of different materials, possess a variety of properties derived from each material. This has brought challenges in both analytical and numerical studies in thermal conduction which is of significant importance for thermoelastic problems. Therefore, a unified and effective approach would be desirable. The present study makes a first attempt to determining the analytical symplectic eigen solution for steady-state thermal conduction problem of multi-material crack. Based on the obtained symplectic eigen solution (including higher order expanding eigen solution terms), a new symplectic analytical singular element (SASE) for numerical modeling is constructed. It is concluded that composite structures composed of multi-material with complex geometric shapes can be modeled by the developed method, and the generalized flux intensity factors (GFIFs) can be solved accurately and efficiently.

A Fast Eigen Solution for Homogeneous Quadratic Minimization With at Most Three Constraints

We propose an eigenvalue based technique to solve the Homogeneous Quadratic Constrained Quadratic Programming problem (HQCQP) with at most 3 constraints which arise in many signal processing problems. Semi-Definite Relaxation (SDR) is the only known approach and is computationally intensive. We study the performance of the proposed fast eigen approach through simulations in the context of MIMO relays and show that the solution converges to the solution obtained using the SDR approach with significant reduction in complexity.

Adaptive algorithms for maximizing overall stock return

Deciding which stocks to purchase and how to optimally allocate the total investment among them is a nontrivial task for every investor. In this article, we propose two adaptive techniques that would provide an optimal allocation maximizing the return over the investment period. The first approach is the adaptive power method (PM) which is a modification of the proper orthogonal decomposition method. The adaptive PM uses only the currently available information to compute the optimal allocations, yet its long-term solution approaches the dominant eigen solution, even though that solution would require having a priori knowledge of all stocks’ performance. The second approach is derived from the well-known Least Mean Square (LMS) method, where the optimal allocation can be computed by adaptively steering the overall return toward a desired value. The experimental results have indicated promising gains even when the general market trend is downward.

WYD method for an eigen solution of coupled problems

Designing efficient and stable algorithm for finding the eigenvalues and eigenvectors is very important from the static as well as the dynamic aspect in coupled problems. Modal analysis requires first few significant eigenvectors and eigenvalues while direct integration requires the highest value to ascertain the length of the time step that satisfies the stability condition. The paper first presents the modification of the well known WYD method for a solution of single field problems: an efficient and numerically stable algorithm for computing eigenvalues and the corresponding eigenvectors. The modification is based on the special choice of the starting vector. The starting vector is the static solution of displacements for the applied load, defined as the product of the mass matrix and the unit displacement vector. The starting vector is very close to the theoretical solution, which is important in cases of small subspaces. Additionally, the paper briefly presents the adopted formulation for solving the fluid-structure coupled systems problems which is based on a separate solution for each field. Individual fields (fluid and structure) are solved independently, taking in consideration the interaction information transfer between them at every stage of the iterative solution process. The assessment of eigenvalues and eigenvectors for multiple fields is also presented. This eigen problem is more complicated than the one for the ordinary structural analysis, as the formulation produces non-symmetrical matrices. Finally, a numerical example for the eigen solution coupled fluidstructure problem is presented to show the efficiency and the accuracy of the developed algorithm.

A study of wind effect on damping and frequency of a long span cable-stayed bridge from rational function approximation of self-excited forces

This paper presents an aeroelastic analysis procedure to highlight the influence of wind velocity on the structural damping and frequency of a long span cable-stayed bridge. Frequency dependent self-excited forces in terms of flutter derivatives are expressed as continuous functions using rational function approximation technique. The aeroelastically modified structural equation of motion is expressed in terms of frequency independent modal state-space parameters. The modal logarithmic dampings and frequencies corresponding to a particular wind speed are then determined from the eigen solution of the state matrix.

A study of wind effect on damping and frequency of a long span cable-stayed bridge from rational function approximation of self-excited forces

This paper presents an aeroelastic analysis procedure to highlight the influence of wind velocity on the structural damping and frequency of a long span cable-stayed bridge. Frequency dependent self-excited forces in terms of flutter derivatives are expressed as continuous functions using rational function approximation technique. The aeroelastically modified structural equation of motion is expressed in terms of frequency independent modal state-space parameters. The modal logarithmic dampings and frequencies corresponding to a particular wind speed are then determined from the eigen solution of the state matrix.

액체추진 로켓의 포고 불안정성 해석과 제어에 관한 연구

Pogo is the instability resulting from the interaction between rocket structure and propulsion system of liquid propellant rocket. The coupling of structure and propulsion system can lead to severe problem in rocket. For the analysis of pogo, a time-invariant linearized mathematical model is developed for a selected flight time. Propulsion system is modeled using element representations for each components. Rocket structure is modeled using FEM. Form the results of modal analysis of structure, the behavior of structure can be represented. System equations for coupling structure and propulsion system are composed. The stability in obtained by the eigen solution of system matrix. The optimization of the design variables such as size, place of accumulator for suppressing pogo instability in carried out. This article of study can be used to determine the degree of stability, and guide the design of pogo suppression system.