Discrete Gradient Research Articles

Properties and practicability of convergence-guaranteed optimization methods derived from weak discrete gradients

The ordinary differential equation (ODE) models of optimization methods allow for concise proofs of convergence rates through discussions based on Lyapunov functions. The weak discrete gradient (wDG) framework discretizes ODEs while preserving the properties of convergence, serving as a foundation for deriving optimization methods. Although various optimization methods have been derived through wDG, their properties and practical applicability remain underexplored. Hence, this study elucidates these aspects through numerical experiments. Particularly, although wDG yields several implicit methods, we highlight the potential utility of these methods in scenarios where the objective function incorporates a regularization term.

A novel directly energy-preserving method for charged particle dynamics

In this paper, we apply the coordinate increment discrete gradient (CIDG) method to solve the Lorentz force system which can be written as a non-canonical Hamiltonian system. Then we can obtain a new energy-preserving CIDG-I method for the system. The CIDG-I method can combine with its adjoint method CIDG-II which is also a energy-preserving method to form a new method, namely CIDG-C method. The CIDG-C method is symmetrical and can conserve the Hamiltonian energy directly and exactly. With comparison to the well-used Boris method, numerical experiments indicate that the CIDG-C method holds advantage over the Boris method in terms of energy-conserving.

Numerical analysis of a mixed-dimensional poromechanical model with frictionless contact at matrix–fracture interfaces

We present a complete numerical analysis for a general discretization of a coupled flow–mechanics model in fractured porous media, considering single-phase flows and including frictionless contact at matrix–fracture interfaces, as well as nonlinear poromechanical coupling. Fractures are described as planar surfaces, yielding the so-called mixed- or hybrid-dimensional models. Small displacements and a linear elastic behavior are considered for the matrix. The model accounts for discontinuous fluid pressures at matrix–fracture interfaces in order to cover a wide range of normal fracture conductivities. The numerical analysis is carried out in the Gradient Discretization framework (see J. Droniou, R. Eymard, T. Gallouët, C. Guichard, and R. Herbin [The gradient discretisation method, Springer, Cham, 2018]), encompassing a large family of conforming and nonconforming discretizations. The convergence result also yields, as a by-product, the existence of a weak solution to the continuous model. A numerical experiment in 2D is presented to support the obtained result, employing a Hybrid Finite Volume scheme for the flow and second-order finite elements ( P 2 \mathbb {P}_2 ) for the mechanical displacement coupled with face-wise constant ( P 0 \mathbb P_0 ) Lagrange multipliers on fractures, representing normal stresses, to discretize the contact conditions.

The lowest-order weak Galerkin finite element method for linear elasticity problems on convex polygonal grids

This paper presents the lowest-order weak Galerkin finite element method for linear elasticity problems on the convex polygonal meshes. This method uses piecewise constant vector-valued spaces on element interiors and edges. The discrete weak gradient space introduced by this paper is the matrix version of CW0 space. The discrete weak divergence space is piecewise constant space on each element. This method is simple, efficient, stabilizer-free and symmetric positive-definite. The optimal error estimates in discrete H1 and L2 norms are presented. Numerical results are given to demonstrate the efficiency of algorithm and the locking-free property.

Enhanced Efficiency in Permanent Magnet Synchronous Motor Drive Systems with MEPA Control Based on an Improved Iron Loss Model

This paper investigates the efficiency optimization problem in the drive system of Permanent Magnet Synchronous Motors (PMSM) based on the iron loss resistance model. Initially, an efficiency calculation method is proposed, utilizing the direct current (DC) side power based on the iron loss motor model as the input power. This approach determines the system’s input power by measuring DC side voltage and current, accounting for the impacts of temperature, magnetic saturation, and inverter nonlinearity. It avoids the need for intricate nonlinear loss modeling of the inverter, thereby simplifying the computational requirements for system efficiency. Simultaneously, the d-q axis equivalent equations based on the iron loss motor model are employed to calculate the system’s output power, enhancing the accuracy of the system efficiency calculation. An improved discrete gradient descent algorithm is presented based on this system efficiency calculation model, accelerating the search for the optimal current angle and improving its accuracy. In comparison to existing methods, this approach exhibits adaptive step size and provides more precise and higher efficiency calculations, resulting in faster search speeds. Experimental and simulation evaluations are conducted to assess the effectiveness of the proposed methods.

A NEW MULTI-STEP BDF ENERGY STABLE TECHNIQUE FOR THE EXTENDED FISHER-KOLMOGOROV EQUATION

The multi-step backward difference formulas of order k (BDF-k) for 3 ≤ k ≤ 5 are proposed for solving the extended Fisher–Kolmogorov equation. Based upon the careful discrete gradient structures of the BDF-k formulas, the suggested numerical schemes are proved to preserve the energy dissipation laws at the discrete levels. The maximum norm priori estimate of the numerical solution is established by means of the energy stable property. With the help of discrete orthogonal convolution kernels techniques, the L2 norm error estimates of the implicit BDF-k schemes are established. Several numerical experiments are included to illustrate our theoretical results.

A variant of the discrete gradient method for the solution of the semilinear wave equation under different boundary conditions

In this paper, energy-preserving schemes based upon the discrete gradient are used to numerically solve the semilinear wave equation under periodic boundary conditions, Dirichlet boundary conditions and Neumann boundary conditions. Both the integral form and the evolutionary behaviour of the energy depend on the associated boundary conditions. The wave equations are semi-discretized in different manners for different boundary conditions such that the Hamiltonian functions of the resulting semi-discrete Hamiltonian systems are the discrete analogue of the original energy. Furthermore, the Hamiltonian functions of the resulting semi-discrete systems have similar evolutionary behaviours as the original energy. It is noted that the semi-discrete system exhibits an oscillatory structure. Subsequently, the semi-discrete approximation of the energy evolution as well as the oscillatory structure is passed to the fully-discrete problem by applying extended discrete gradient formula in the temporal direction. Numerical experiments are carried out to show the effectiveness of the new schemes.

Energy dissipation law of the variable time-step fractional BDF2 scheme for the time fractional molecular beam epitaxial model

In this work, we take a consideration of the time fractional molecular beam epitaxial(MBE) models. The variable time-step BDF2 methods are proposed for time fractional MBE models in order to obtain the high-order accuracy in time since the low regularity occurring in the initial state. By virtue of discrete gradient structures, energy dissipation laws of the schemes for time fractional MBE model with and without slope selection are proved, respectively. Besides, the unique solvabilities of two schemes are also guaranteed. The discrete modified energy and the corresponding energy dissipation law are asymptotically compatible with the associated discrete energy and the energy dissipation law of the variable time-step BDF2 method for the classical MBE model. Numerical experiments with the time adaptive strategies are provided to verify the effectiveness of the schemes.

A discrete adjoint gradient approach for equality and inequality constraints in dynamics

The optimization of multibody systems requires accurate and efficient methods for sensitivity analysis. The adjoint method is probably the most efficient way to analyze sensitivities, especially for optimization problems with numerous optimization variables. This paper discusses sensitivity analysis for dynamic systems in gradient-based optimization problems. A discrete adjoint gradient approach is presented to compute sensitivities of equality and inequality constraints in dynamic simulations. The constraints are combined with the dynamic system equations, and the sensitivities are computed straightforwardly by solving discrete adjoint algebraic equations. The computation of these discrete adjoint gradients can be easily adapted to deal with different time integrators. This paper demonstrates discrete adjoint gradients for two different time-integration schemes and highlights efficiency and easy applicability. The proposed approach is particularly suitable for problems involving large-scale models or high-dimensional optimization spaces, where the computational effort of computing gradients by finite differences can be enormous. Three examples are investigated to validate the proposed discrete adjoint gradient approach. The sensitivity analysis of an academic example discusses the role of discrete adjoint variables. The energy optimal control problem of a nonlinear spring pendulum is analyzed to discuss the efficiency of the proposed approach. In addition, a flexible multibody system is investigated in a combined optimal control and design optimization problem. The combined optimization provides the best possible mechanical structure regarding an optimal control problem within one optimization.

Full weak Galerkin finite element discretizations for poroelasticity problems in the primal formulation

This paper develops novel finite element solvers for linear poroelasticity problems on quadrilateral meshes. These solvers are based on the primal formulations of linear elasticity and Darcy flow. Specifically, the fluid pressure and solid displacement are approximated by scalar- or vector-valued polynomials of degree k≥0 separately in element interiors and on edges. The discrete weak gradients of these shape functions are established in the broken (vector- or matrix-version) Arbogast–Correa spaces for approximations of the classical gradients in the variational forms. These weak Galerkin spatial discretizations are combined with the implicit Euler or Crank–Nicolson temporal discretizations to develop locking-free numerical solvers that have optimal order (k+1) convergence rates in pressure, velocity, displacement, stress, and dilation. Rigorous analysis is presented and illustrated by numerical experiments on popular test cases.

Energy-preserving schemes for conservative PDEs based on periodic quasi-interpolation methods

In this paper, we present a novel energy-preserving scheme for solving time-dependent partial differential equations with periodic solutions on non-uniform grids. The proposed scheme combines the periodic quasi-interpolation approaches with the discrete gradient method. One of the main advantages of our method is its ability to ensure energy conservation without relying on the requirement of an anti-symmetric differentiation matrix. We rigorously verify the energy conservation properties of the full-discrete scheme, as well as establish the convergence results. Furthermore, we extend the capabilities of our method by incorporating an adaptive strategy, allowing for the design of an adaptive energy-preserving scheme. This adaptive feature enhances the accuracy and efficiency of the numerical solution. To demonstrate the effectiveness and energy-preserving ability of the proposed scheme, we present several numerical examples. These examples showcase the superior accuracy and robustness of our method.

Discrete gradients in short-range molecular dynamics simulations

Discrete gradients (DG) or more exactly discrete gradient methods are time integration schemes that are custom-built to preserve first integrals or Lyapunov functions of a given ordinary differential equation (ODE). In conservative molecular dynamics (MD) simulations, the energy of the system is constant and therefore a first integral of motion. Hence, discrete gradient methods seem to be a natural choice as an integration scheme in conservative molecular dynamics simulations.

Fréchet Discrete Gradient and Hessian Operators on Infinite-Dimensional Spaces

Benefiting from the notion of Fréchet derivatives, we define Fréchet discrete operators, such as gradient and Hessian, on infinite-dimensional spaces. The Fréchet discrete gradient expands upon the concept of the discrete gradient of Gonzalez (1996) for finite-dimensional spaces. The Fréchet discrete Hessian elevates the property to second-order representations of the Fréchet derivative. By leveraging these operators, we offer an initial exploration of discrete gradient methods for convex optimization in infinite-dimensional spaces. Under mild conditions on the objective functional, we establish the convergence of any sequence generated by the proposed Fréchet discrete gradient method, regardless of the choice of the finite learning rate.

Passive-guaranteed modeling and simulation of a finite element nonlinear string model

This paper proposes to solve the dynamics of the Kirchhoff-Carrier nonlinear string model using the finite elements method. In order to ensure the power balance of the resulting finite dimensional model it is rewritten in the Port-Hamiltonian System (PHS) formalism. Using a discrete gradient and a quadratization of the Hamiltonian, an explicit power-preserving numerical scheme is proposed. Results of simulation are presented.

Expected Integral Discrete Gradient: Diagnosing Autonomous Driving Model

Expected Integral Discrete Gradient: Diagnosing Autonomous Driving Model

Derivative-free discrete gradient methods

Derivative-free discrete gradient methods

Localized Evaluation for Constructing Discrete Vector Fields.

Topological abstractions offer a method to summarize the behavior of vector fields, but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach, which constructs a collection of pairs of simplices in the input mesh that satisfies criteria introduced by Forman's discrete Morse theory. While numerous approaches exist to compute pairs in the restricted case of the gradient of a scalar field, state-of-the-art algorithms for the general case of vector fields require expensive optimization procedures. This paper introduces a fast, novel approach for pairing simplices of two-dimensional, triangulated vector fields that do not vary in time. The key insight of our approach is that we can employ a local evaluation, inspired by the approach used to construct a discrete gradient field, where every simplex in a mesh is considered by no more than one of its vertices. Specifically, we observe that for any edge in the input mesh, we can uniquely assign an outward direction of flow. We can further expand this consistent notion of outward flow at each vertex, which corresponds to the concept of a downhill flow in the case of scalar fields. Working with outward flow enables a linear-time algorithm that processes the (outward) neighborhoods of each vertex one-by-one, similar to the approach used for scalar fields. We couple our approach to constructing discrete vector fields with a method to extract, simplify, and visualize topological features. Empirical results on analytic and simulation data demonstrate drastic improvements in running time, produce features similar to the current state-of-the-art, and show the application of simplification to large, complex flows.

Multiresolution approximation for shallow water equations using summation-by-parts finite differences

Abstract We present spatial approximation for shallow water equations on a mesh of multiple rectangular blocks with different resolution in Cartesian geometry. The approximation is based on finite-difference operators that fulfill Summation By Parts (SBP) property – a discrete analogue of integration by parts. The solution continuity conditions between mesh blocks are imposed in a weak form using Simultaneous Approximation Terms (SAT) method.We show that the resulting discrete divergence and gradient operators are anti-conjugate. The important consequences are the discrete analogues for mass and energy conservation laws along with the proof of stability for linearized equations. The numerical shallow water equations model based on the presented spatial approximation is tested using problems with meteorological context. Test results prove high-order accuracy of SBP-SAT discretization. The interfaces between mesh blocks of different resolution produce no significant noise. The local mesh refinement is shown to have positive effect on the solution both locally inside the refined region and globally in the dynamically coupled areas.

Asymptotically compatible energy of two variable-step fractional BDF2 schemes for the time fractional Allen–Cahn model

In this paper, the variable-step fractional BDF2 (FBDF2) formula is applied to solve the time fractional Allen–Cahn (TFAC) model. By the discrete gradient structure of the FBDF2 operator and the local–nonlocal splitting technique, the asymptotically compatible energy of the nonlinear fully implicit FBDF2 scheme and the linearly FBDF2-SAV scheme by the scalar auxiliary variable technique are established for the TFAC model. Finally, several numerical experiments are provided to verify the theoretical results and compare the effectiveness of two numerical schemes.

Vertex-Centered Mixed Finite Element–Finite Volume scheme for 2D anisotropic hybrid mesh adaptation

The scope of this paper is to propose an appropriate numerical strategy to extend the Mixed Finite Element–Finite Volume (MEV) scheme to adapted meshes composed of both triangular and quadrangular elements, called hybrid meshes (Kallinderis and Kavouklis, 2005; Ito et al., 2013) (also named mixed-element meshes (Marcum and Gaither, 1999; Mavriplis, 2000)). Convective, diffusive and source terms require specific gradient discretization to maintain a global second-order accuracy. The major contributions of this paper concern the specification of these gradients, the convective fluxes’ discretization and an innovative framework for anisotropic metric-based adaptation on hybrid meshes. Particularly, the V4 scheme, traditionally developed for triangular elements, is extended to quadrilateral elements. This method maintains the same order of accuracy as the baseline version when upwind/downwind quadrilateral elements are aligned with the attached edge. Viscous fluxes are discretized using the APproximated Finite Element (APFE) method (Puigt et al., 2010), which is second-order accurate on regular quadrilaterals. Nodal gradients, which are considered at boundaries and for source terms, are discretized with a generalization of Clement’s L2-operator on hybrid meshes. The mesh adaptation relies on a specific metric gradation process to favor the clustering of structured elements at wall boundaries. The proposed scheme is validated on a set of CFD simulations.