Discrete Variational Principle Research Articles

Variational discretizations of ideal magnetohydrodynamics in smooth regime using structure-preserving finite elements

We propose a new class of finite element approximations to ideal compressible magnetohydrodynamic equations in smooth regime. Following variational approaches developed for fluid models in the last decade, our discretizations are built via a discrete variational principle mimicking the continuous Euler-Poincaré principle, and to further exploit the geometrical structure of the problem, vector fields are represented by their action as Lie derivatives on differential forms of any degree. The resulting semi-discrete approximations are shown to conserve the total mass, entropy and energy of the solutions for a wide class of finite element approximations. In addition, the divergence-free nature of the magnetic field is preserved in a pointwise sense and a time discretization is proposed, preserving those invariants and giving a reversible scheme at the fully discrete level. Numerical simulations are conducted using spline elements to verify the accuracy of our approach and its ability to preserve the invariants for several test problems.

Hamel's field variational integrator for simulating dynamics of thin-walled geometrically exact beams with warping effects

Hamel's field variational integrator for simulating dynamics of thin-walled geometrically exact beams with warping effects

A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion

In this paper, we propose an efficient numerical method for a comprehensive infection model that is formulated by a system of nonlinear coupling advection-diffusion-reaction equations. Using some subtle mixed explicit-implicit treatments, we construct a linearized and decoupled discrete scheme. Moreover, the proposed scheme is capable of preserving the positivity of variables, which is an essential requirement of the model under consideration. The proposed scheme uses the cell-centered finite difference method for the spatial discretization, and thus, it is easy to implement. The diffusion terms are treated implicitly to improve the robustness of the scheme. A semi-implicit upwind approach is proposed to discretize the advection terms, and a distinctive feature of the resulting scheme is to preserve the positivity of variables without any restriction on the spatial mesh size and time step size. We rigorously prove the unique existence of discrete solutions and positivity-preserving property of the proposed scheme without requirements for the mesh size and time step size. It is worthwhile to note that these properties are proved using the discrete variational principles rather than the conventional approaches of matrix analysis. Numerical results are also provided to assess the performance of the proposed scheme.

Sound synthesis of a 3D nonlinear string using a covariant Lie group integrator of a geometrically exact beam model

Sound synthesis of a 3D nonlinear string using a covariant Lie group integrator of a geometrically exact beam model

High-order integrators for Lagrangian systems on homogeneous spaces via nonholonomic mechanics

High-order integrators for Lagrangian systems on homogeneous spaces via nonholonomic mechanics

A novel four-field mixed FE approximation for Kirchhoff rods using Cartan’s moving frames

A novel four-field mixed FE approximation for Kirchhoff rods using Cartan’s moving frames

Backward error analysis for variational discretisations of PDEs

<p style='text-indent:20px;'>In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby 'modified' equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation.</p>

Optimal control simulations of two-finger grasps

Optimal control simulations of two-finger grasps

Geometric electrostatic particle-in-cell algorithm on unstructured meshes

We present a geometric particle-in-cell (PIC) algorithm on unstructured meshes for studying electrostatic perturbations with frequency lower than electron gyrofrequency in magnetized plasmas. In this method, ions are treated as fully kinetic particles and electrons are described by the adiabatic response. The PIC method is derived from a discrete variational principle on unstructured meshes. To preserve the geometric structure of the system, the discrete variational principle requires that the electric field is interpolated using Whitney 1-forms, the charge is deposited using Whitney 0-forms and the electric field is computed by discrete exterior calculus. The algorithm has been applied to study the ion Bernstein wave (IBW) in two-dimensional magnetized plasmas. The simulated dispersion relations of the IBW in a rectangular region agree well with theoretical results. In a two-dimensional circular region with fixed boundary condition, the spectrum and eigenmode structures of the IBW are obtained from simulations. We compare the energy conservation property of the geometric PIC algorithm derived from the discrete variational principle with that of previous PIC methods on unstructured meshes. The comparison shows that the new PIC algorithm significantly improves the energy conservation property.

A combinatorial Yamabe problem on two and three dimensional manifolds

In this paper, we define a new discrete curvature on two and three dimensional triangulated manifolds, which is a modification of the well-known discrete curvature on these manifolds. The new definition is more natural and respects the scaling exactly the same way as Gauss curvature does. Moreover, the new discrete curvature can be used to approximate the Gauss curvature on surfaces. Then we study the corresponding constant curvature problem, which is called the combinatorial Yamabe problem, by the corresponding combinatorial versions of Ricci flow and Calabi flow for surfaces and Yamabe flow for 3-dimensional manifolds. The basic tools are the discrete maximal principle and variational principle.

Dynamic modeling of soft continuum manipulators using lie group variational integration.

This paper presents the derivation and experimental validation of algorithms for modeling and estimation of soft continuum manipulators using Lie group variational integration. Existing approaches are generally limited to static and quasi-static analyses, and are not sufficiently validated for dynamic motion. However, in several applications, models need to consider the dynamical behavior of the continuum manipulators. The proposed modeling and estimation formulation is obtained from a discrete variational principle, and therefore grants outstanding conservation properties to the continuum mechanical model. The main contribution of this article is the experimental validation of the dynamic model of soft continuum manipulators, including external torques and forces (e.g., generated by magnetic fields, friction, and the gravity), by carrying out different experiments with metal rods and polymer-based soft rods. To consider dissipative forces in the validation process, distributed estimation filters are proposed. The experimental and numerical tests also illustrate the algorithm’s performance on a magnetically-actuated soft continuum manipulator. The model demonstrates good agreement with dynamic experiments in estimating the tip position of a Polydimethylsiloxane (PDMS) rod. The experimental results show an average absolute error and maximum error in tip position estimation of 0.13 mm and 0.58 mm, respectively, for a manipulator length of 60.55 mm.

Symplectic Instantaneous Optimal Control of Deployable Structures Driven by Sliding Cable Actuators

A symplectic instantaneous optimal control (IOC) method is presented for compression/deployment of structures driven by sliding cable actuators. First, the motion of these active sliding cables is regarded as system kinematic constraints from the view of multibody dynamics, and a more general structural deployment control system can be established by differential-algebraic equations (DAEs). Mathematically, the system can be depicted as a constrained nonlinear optimal control problem for DAE systems. Then, based on the Lagrange–d’Alembert principle and the discrete variational principle, a symplectic discretization format can be constructed. The original continuous control problem will be approximated into a series of constrained IOC problems at every time slot, and to satisfy the inequality constraints of input saturation, the linear complementarity problem can finally be derived for solving these IOC problems. The proposed method provides a novel control strategy for addressing more general structure deployment control problems with fewer actuators. Structures can be quickly compressed/deployed to the target configuration, and the residual vibration can be minimized at the same time. The effectiveness and universality of the proposed method are further illustrated by numerical simulations of a telescopic tensegrity manipulator and a deployable tensegrity antenna.

Stabilized Energy Factorization Approach for Allen–Cahn Equation with Logarithmic Flory–Huggins Potential

The Allen--Cahn equation is one of fundamental equations of phase-field models, while the logarithmic Flory--Huggins potential is one of the most useful energy potentials in various phase-field models. In this paper, we consider numerical schemes for solving the Allen--Cahn equation with logarithmic Flory--Huggins potential. The main challenge is how to design efficient numerical schemes that preserve the maximum principle and energy dissipation law due to the strong nonlinearity of the energy potential function. We propose a novel energy factorization approach with the stability technique, which is called stabilized energy factorization approach, to deal with the Flory--Huggins potential. One advantage of the proposed approach is that all nonlinear terms can be treated semi-implicitly and the resultant numerical scheme is purely linear and easy to implement. Moreover, the discrete maximum principle and unconditional energy stability of the proposed scheme are rigorously proved using the discrete variational principle. Numerical results are presented to demonstrate the stability and effectiveness of the proposed scheme.

Preservation of adiabatic invariants for disturbed Hamiltonian systems under variational discretization

The perturbation to conformal invariance and the numerical algorithm of Hamiltonian systems with disturbed forces under variational discretization are studied in this paper. Based on the discrete difference variational principles, the discrete Hamiltonian equations (variational integrators) for dynamical systems are obtained in the undisturbed and the disturbed cases, respectively. The determining equations of perturbation to conformal invariance are established for disturbed Hamiltonian systems. The exact invariants of Noether type led by conformal invariance for an undisturbed Hamiltonian system are derived. For disturbed discrete Hamiltonian systems, the condition of perturbation to conformal invariance leading to adiabatic invariants is proposed. Two examples are considered: a simple harmonic oscillator and the Kepler problem. The dynamical analysis is given by using the numerical results.

A Symplectic Instantaneous Optimal Control for Robot Trajectory Tracking With Differential-Algebraic Equation Models

Robot trajectory tracking control based on differential-algebraic equation (DAE) models is still a thorny issue, because the DAEs of such systems are inherently complex and unstable, such as the high-index problem. In this paper, a symplectic instantaneous optimal control (IOC) method for robot trajectory tracking with input saturation based on controlled DAEs is proposed. Based on the discrete variational principle and the canonical transformation, a symplectic discretization form for the controlled DAEs is first constructed. Then, the continuous trajectory tracking problem is approximated for a series of IOC problems at every time step, and the linear complementarity problem (LCP) can be derived for solving the IOC problems. Finally, the control inputs can be obtained by solving the corresponding standard LCP. The proposed method provides a unified framework for solving the trajectory tracking control problems of robot multibody dynamic systems. Numerical simulations and virtual experiments are conducted to verify the robustness and the efficiency of the proposed method, i.e., the input saturation constraints are satisfied at the discrete time points, and high accuracy tracking control results can be obtained at low computational cost.

Optimization based muscle wrapping in biomechanical multibody simulations

AbstractIn musculoskeletal simulations muscle lengths and muscle force directions imply the characteristics of muscles acting around joints. Typically, the anatomical structure of the human body forces muscles to wrap around bones and neighboring tissue, thus most muscle paths cannot be represented adequately as straight lines. Therefore, biomechanical simulations require methods to compute musculotendon paths, their lengths, and their rates of length change to determine the muscle forces. This work focuses on a mechanical analogue to find the shortest path on general smooth surfaces, using a discrete variational principle. In this context, the geodesic path is reinterpreted as the constrained, force‐free motion of a particle in n dimensions. The muscle path is then a G1‐continuous combination of geodesics on adjacent obstacle surfaces [1,3,4].

Variational integrators for reduced field equations

In the reduction of field theories in principal $G$-bundles, when a subgroup $H\subset G$ acts by symmetries of the Lagrangian, each of the $H$-reduced unknown fields decomposes as a flat principal connection and a parallel H-structure. A suitable variational principle with differential constraints on such fields leads to necessary criticality conditions known as Euler-Poincare equations. We model constrained discrete variational theories on a simplicial complex and generate from the smooth theory, in a covariant way, a discrete variational formulation of $H$-reduced field theories. Critical fields in this formulation are characterized by a corresponding discrete version of Euler-Poincare equations. We present a numerical integration algorithm for discrete Euler-Poincare equations, that extends integration algorithms for Euler-Poincare equations in mechanics to the case of field theories and, also, extends integration algorithms for Euler-Lagrange equations in discrete field theories to the case of variational principles with constraints. For regular reduced discrete Lagrangians, this algorithm allows to univocally propagate initial condition data, on an initial condition band, into a solution of the corresponding equations for the discrete variational principle.

Efficient numerical schemes for Chan-Vese active contour models in image segmentation

In this paper, we introduce multi-symplectic Lagrangian variational integrators for solving Chan-Vese active contour models in image segmentation. Energy functionals are discretized firstly, and numerical schemes are derived from discrete Euler-Lagrange equations based on discrete variational principle. Lagrangian variational integrators preserve native differential structure-multi-symplecticity, that makes the numerical methods have a satisfied behavior. Experiments are performed on the benchmark images from literature. We further evaluated the methods in a segmentation database containing 1023 images. It shows that the proposed numerical schemes attain relatively faster convergence rates and better segmentation accuracy. Comparisons with the standard explicit Euler method of the original Chan-Vese model and other fast numerical optimization methods show that the proposed methods have better stability, higher accuracy, and are more robust when dealing with a large number of pictures. This study provides an example for further research to improve the performance of other existing image segmentation methods based on active contour models.

Galerkin‐based energy‐momentum time‐stepping schemes for anisotropic hyperelastic materials

AbstractThis paper is motivated by dynamic simulations of fiber‐reinforced materials in light‐weight structures, and has two goals. First of all, the introduction of energy‐momentum schemes for nonlinear anisotropic materials, based on GALERKIN approximations in space and time, assumed strain approximations in time and superimposed algorithmic stress fields (compare [1]). Second, to show a variationally consistent design of energy‐momentum schemes using a differential variational principle. We develop a discrete variational principle leading to energy‐momentum schemes as discrete EULER‐LAGRANGE equations. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Variational integrators for reduced magnetohydrodynamics

Variational integrators for reduced magnetohydrodynamics