Geometric formulations and variational integrators of discrete autonomous Birkhoff systems

The variational integrators of autonomous Birkhoff systems are obtained by the discrete variational principle. The geometric structure of the discrete autonomous Birkhoff system is formulated. The discretization of mathematical pendulum shows that the discrete variational method is as effective as symplectic scheme for the autonomous Birkhoff systems.

Chebyshev spectral variational integrator and applications

The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties.According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points,the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator(SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system.

GENERALIZED VARIATIONAL PRINCIPLES OF THE VISCOELASTIC BODY WITH VOIDS AND THEIR APPLICATIONS

From the Boltzmann's constitutive law of viscoelastic materials and the linear theory of elastic materials with voids, a constitutive model of generalized force fields for viscoelastic solids with voids was given. By using the variational integral method, the convolution-type functional was given and the corresponding generalized variational principles and potential energy principle of viscoelastic solids with voids were presented. It can be shown that the variational principles correspond to the differential equations and the initial and boundary conditions of viscoelastic body with voids. As an application, a generalized variational principle of viscoelastic Timoshenko beams with damage was obtained which corresponds to the differential equations of generalized motion and the initial and boundary conditions of beams. The variational principles provide a way for solving problems of viscoelastic solids with voids.

Symmetries and variational calculation of discrete Hamiltonian systems

We present a numerical simulation method of Noether and Lie symmetries for discrete Hamiltonian systems. The Noether and Lie symmetries for the systems are proposed by investigating the invariance properties of discrete Lagrangian in phase space. The numerical calculations of a two-degree-of-freedom nonlinear harmonic oscillator show that the difference discrete variational method preserves the exactness and the invariant quantity.

New way to construct high order Hamiltonian variational integrators

This paper develops a new approach to construct variational integrators. A simplified unconventional Hamilton＇s variational principle corresponding to initial value problems is proposed, which is convenient for applications. The displacement and mo- mentum are approximated with the same Lagrange interpolation. After the numerical integration and variational operation, the original problems are expressed as algebraic equations with the displacement and momentum at the interpolation points as unknown variables. Some particular variational integrators are derived. An optimal scheme of choosing initial values for the Newton-Raphson method is presented for the nonlinear dynamic system. In addition, specific examples show that the proposed integrators are symplectic when the interpolation point coincides with the numerical integration point, and both are Gaussian quadrature points. Meanwhile, compared with the same order symplectic Runge-Kutta methods, although the accuracy of the two methods is almost the same, the proposed integrators are much simpler and less computationally expensive.

Dynamic modeling of spacecraft solar panels deployment with Lie group variational integrator

The spacecraft with multistage solar panels have nonlinear coupling between attitudes of central body and solar panels, especially the rotation of central body is considered in space. The dynamics model is based for dynamics analysis and control, and the multistage solar panels means the dynamics modeling will be very complex. In this research, the Lie group variational integrator method is introduced, and the dynamics model of spacecraft with solar panels that connects together by flexible joints is built. The most obvious character of this method is that the attitudes of central body and solar panels are all described by three-dimensional attitude matrix. The dynamics models of spacecraft with one and three solar panels are established and simulated. The study shows Lie group variational integrator method avoids parameters coupling and effectively reduces difficulty of modeling. The obtained continuous dynamics model based on Lie group is a set of ordinary differential equations and equivalent with traditional dynamics model that offers a basis for the geometry control.

Multi-symplectic variational integrators for nonlinear Schrdinger equations with variable coefficients

In this paper, we propose a variational integrator for nonlinear Schrodinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrodinger equations with variable coefficients, cubic nonlinear Schrodinger equations and Gross-Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space.

VARIATIONAL PRINCIPLES OF FLUID FULLFILLED ELASTIC SOLIDS

The generalized variational principles of isothermal quasi-static fluid full-filled elastic solids are established by using Variational Integral Method. Then by introducing constraints, several kinds of variational principles are worked out, including five-field variable, four-field variable, three-field variable and two-field variable formulations. Some new variational principles are presented besides the principles noted in the previous works. Based on variational principles, finite element models can be set up.

On the Inverse Problem in calculus of Variations

The inverse problem in calculus of variation is studied. By introducing a newconcept called Varialional Integral, a new method to systematically study the inverseproblem in calculus of rariations is given. Using this new method to the elastodynamicsand hydrodynamics of viscous fhuids some kinds of variaiional principles andgeneralized variational prineiples are obtained respectively.

Influence of Contractility on Myocardial Ultrasonic Integrated Backscatter and Cyclic Variation in Integrated Backscatter

To evaluate the effects of left ventricular contractility on the changes of aver age image intensity (AII) of the myocardial integrated backscatter (IB) and cyclic variation in IB (CVIB), 7 adult mongrel dogs were studied. The magnitude of AII and CVIB were measured from myocardial IB carves before and after dobuta mine or propranolol infusion. Dobutamine or propranolol did not affect the magnitude of AII (13.8±0.7 vs 14.7±0.5, P >0.05 or 14.3±0.5 vs 14.2±0.4, P >0.05). However, dobutamine produced a significant increase in the magnitude of CVIB (6.8±0.3 vs 9.5±0.6, P <0.001) and propranolol induced significant decrease in the magnitude of CVIB (7.1±0.2 vs 5.2±0.3, P <0.001). The changes of the magnitude of AII and CVIB in the myocardium have been demonstrated to reflect different myocardial physiological and pathological changes respectively. The alteration of contractility did not affect the magnitude of AII but induced significant change in CVIB. The increase of left ventricular contractility res ulted in a significant rise of the magnitude of CVIB and the decrease of left ventricular contractility resulted in a significant fall of the magnitude of CVIB.

Regularity for minimizing sequences of some variational integrals

This paper deals with regularity properties for minimizing sequences of some integral functionals related to the nonlinear elasticity theory.Under some structural conditions,we derive that the minimizing sequence and the derivatives of the sequences have some regularity properties by using the Ekeland variational principle.

An Asynchronous Variational Integrator for Contact Problems Involving Elastoplastic Solids

Simulations of contact problems involving at least one plastic solid may be costly due to their strong nonlinearity and requirements of stability.In this work,we develop an explicit asynchronous variational integrator(AVI)for inelastic non-frictional contact problems involving a plastic solid.The AVI assigns each element in the mesh an independent time step and updates the solution at the elements and nodes asynchronously.This asynchrony makes the AVI highly efficient in solving such bi-material problems.Taking advantage of the AVI,the constitutive update is locally performed in one element at a time,and contact constraints are also enforced on only one element.The time step of the contact element is subdivided into multiple segments,and the fields are updated accordingly.During a contact event,only one element involving a few degrees of freedom is considered,leading to high efficiency.The proposed formulation is first verified with a pure elastodynamics benchmark and further applied to a contact problem involving an elastoplastic solid with non-associative volumetric hardening.The numerical results indicate that the AVI exhibits excellent energy behaviors and has high computational efficiency.

GIBBS-APPELL’S EQUATIONS OF VARIABLE MASS NONLINEAR NONHOLONOMIC MECHANICAL SYSTEMS

In this paper, the Gibbs-Appell's equations of motion are extended to the most general variable mass nonholonomie mechanical systems. Then the Gibbs-Appell's equations of motion in terms of generalized coordinates or quasi-coordinates and an integral variational principle of variable mass nonlinear nonholonomie mechanical systems are obtained. Finally, an example is given.

High-Order Hamilton's Principle and the Hamilton's Principle of High-Order Lagrangian Function

In this paper, based on the theorem of the high-order velocity energy, integration and variation principle, the high-order Hamilton＇s principle of general holonomic systems is given. Then, three-order Lagrangian equations and four-order Lagrangian equations are obtained from the high-order Hamilton＇s principle. Finally, the Hamilton＇s principle of high-order Lagrangian function is given.

Understanding biological functions through molecular networks

The completion of genome sequences and subsequent high-throughput mapping of molecular networks have allowed us to study biology from the network perspective. Experimental, statistical and mathematical modeling approaches have been employed to study the structure, function and dynamics of molecular networks, and begin to reveal important links of various network properties to the functions of the biological systems. In agreement with these functional links, evolutionary selection of a network is apparently based on the function, rather than directly on the structure of the network. Dynamic modularity is one of the prominent features of molecular networks. Taking advantage of such a feature may simplify network-based biological studies through construction of process-specific modular networks and provide functional and mechanistic insights linking genotypic variations to complex traits or diseases, which is likely to be a key approach in the next wave of understanding complex human diseases. With the development of ready-to-use network analysis and modeling tools the networks approaches will be infused into everyday biological research in the near future.

Error Estimations, Error Computations, and Convergence Rates in FEM for BVPs

This paper presents derivation of a priori error estimates and convergence rates of finite element processes for boundary value problems (BVPs) described by self adjoint, non-self adjoint, and nonlinear differential operators. A posteriori error estimates are discussed in context with local approximations in higher order scalar product spaces. A posteriori error computational framework (without the knowledge of theoretical solution) is presented for all BVPs regardless of the method of approximation employed in constructing the integral form. This enables computations of local errors as well as the global errors in the computed finite element solutions. The two most significant and essential aspects of the research presented in this paper that enable all of the features described above are: 1) ensuring variational consistency of the integral form(s) resulting from the methods of approximation for self adjoint, non-self adjoint, and nonlinear differential operators and 2) choosing local approximations for the elements of a discretization in a subspace of a higher order scalar product space that is minimally conforming, hence ensuring desired global differentiability of the approximations over the discretizations. It is shown that when the theoretical solution of a BVP is analytic, the a priori error estimate (in the asymptotic range, discussed in a later section of the paper) is independent of the method of approximation or the nature of the differential operator provided the resulting integral form is variationally consistent. Thus, the finite element processes utilizing integral forms based on different methods of approximation but resulting in VC integral forms result in the same a priori error estimate and convergence rate. It is shown that a variationally consistent (VC) integral form has best approximation property in some norm, conversely an integral form with best approximation property in some norm is variationally consistent. That is best approximation property of the integral form and the VC of the integral form is equivalent, one cannot exist without the other, hence can be used interchangeably. Dimensional model problems consisting of diffusion equation, convection-diffusion equation, and Burgers equation described by self adjoint, non-self adjoint, and nonlinear differential operators are considered to present extensive numerical studies using Galerkin method with weak form (GM/WF) and least squares process (LSP) to determine computed convergence rates of various error norms and present comparisons with the theoretical convergence rates.

Analytic Feynman Integrals of Functionals in a Banach Algebra Involving the First Variation

This paper deals with the analytic Feynman integral of functionals on a Wiener space. First the authors establish the existence of the analytic Feynman integrals of functionals in a Banach algebra S_α. The authors then obtain a formula for the first variation of integrals. Finally, various analytic Feynman integration formulas involving the first variation are established.

General techniques for constructing variational integrators

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi＇s solution of the Hamilton- Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators We also illustrate these systematic methods for constructing variational integrators with numerical examples.

VARIATIONAL INTEGRATORS FOR HIGHER ORDER DIFFERENTIAL EQUATIONS

We analyze three one parameter families of approximations and show that they are symplectic in Lagrangian sence and can be related to symplectic schemes in Hamiltonian sense by different symplectic mappings. We also give a direct generalization of Veselov variational principle for construction of scheme of higher order differential equations. At last, we present numerical experiments.

Implementation Details for the Phase Field Approaches to Fracture

Phase field description of fracture is a very promising approach for simulating crack initiation, propagation, merging and branching. This method greatly reduces the implementation complexity, compared with discrete descriptions of cracks. In this work, we provide an overview of phase field models for quasistatic and dynamic cases. Afterward, we present useful vectors and matrices for the implementation of this method in two and three dimensions.