Time Discretization Procedure Research Articles

A time-adaptive FE2-approach within the method of vertical lines

FE2-approaches are chosen to incorporate micro-mechanical information into macroscopical computations. Although the approach is applied for several years, there are some open theoretical aspects in the algorithmic description. In this contribution the entire concept is brought into the method of vertical lines to solve initial boundary-value problems based on the homogenization concept under consideration. This requires a clear description of known and unknown quantities in the space and time discretization procedure. Dependent on the constitutive model, systems of algebraic or differential-algebraic equations (DAE) arise. Here, we concentrate on models of evolutionary type. Frequently, the resulting DAE-systems are solved using the Backward-Euler method, which are embedded in the more general class of diagonally, implicit Runge-Kutta methods. After the time discretization the resulting systems of non-linear equations have to be solved. This can be done using various schemes to obtain more efficient computations. Here, the three-level Newton algorithm, the Newton-Raphson-Schur method, numerical differentiation and a Chord-version are derived in the context of FE2 and discussed. Finally, step-size-controlled FE2-problems are solved, highlighting the importance of time-adaptive computations, particularly, in combination with the non-linear solution scheme, where the Multilevel-Newton-Chord method shows high efficiency. This contributes to an additional reduction of the overall computational times.

Well‐posedness for a diffusion–reaction compartmental model simulating the spread of COVID‐19

This paper is concerned with the well‐posedness of a diffusion–reaction system for a susceptible‐exposed‐infected‐recovered (SEIR) mathematical model. This model is written in terms of four nonlinear partial differential equations with nonlinear diffusions, depending on the total amount of the SEIR populations. The model aims at describing the spatio‐temporal spread of the COVID‐19 pandemic and is a variation of the one recently introduced, discussed, and tested in a paper by Viguerie et al (2020). Here, we deal with the mathematical analysis of the resulting Cauchy–Neumann problem: The existence of solutions is proved in a rather general setting, and a suitable time discretization procedure is employed. It is worth mentioning that the uniform boundedness of the discrete solution is shown by carefully exploiting the structure of the system. Uniform estimates and passage to the limit with respect to the time step allow to complete the existence proof. Then, two uniqueness theorems are offered, one in the case of a constant diffusion coefficient and the other for more regular data, in combination with a regularity result for the solutions.

Coordinated Vehicle Platooning with Fixed Routes: Adaptive Time Discretization, Strengthened Formulations and Approximation Algorithms

We consider the coordinated vehicle platooning problem over a road network with time constraints and the routes of vehicles are given. The problem is to coordinate the departure time of each vehicle to enable platoon formation hence maximizing the total fuel saving. We first focus on the case that the routes form a tree. In this case, the flexibility time window of each vehicle admits a unified representation, under which the synchronization of departure time is equivalent to the management of time-window overlapping. Based on this observation, a time-window-overlapping induced mixed-integer linear program formulation has been established which is equivalent to the continuous-time formulation established in the previous work. By imposing an adaptive time-discretization procedure, we can further reformulate the time coordination problem using a vehicle-to-time-bucket assignment induced mixed-integer linear program. Compared to the continuous-time formulation, the assignment formulation is free of big-M coefficients, and it is demonstrated by numerical experiments that the assignment formulation is indeed more effective in computational performance. As an independent interest, three approximation algorithms have been derived with provable competitive ratios under a certain regularity assumptions, which is for the first time on this discrete optimization problem. The solution approach has been extended to general cases in which the graph formed by vehicle routes can have loops, by adding a systematic loop breaking scheme. An insightful discovery is that whether there exists an exact equivalent assignment formulation for a general problem instance has a close connection to the pattern of lattice groups.

Numerical study of the nonlinear anomalous reaction–subdiffusion process arising in the electroanalytical chemistry

This paper presents a meshless method based on the finite difference scheme derived from the local radial basis function (RBF-FD). The algorithm is used for finding the approximate solution of nonlinear anomalous reaction–diffusion models. The time discretization procedure is carried out by means of a weighted discrete scheme covering second-order approximation, while the spatial discretization is accomplished using the RBF-FD. The theoretical discussion validates the stability and convergence of the time-discretized formulation which are analyzed in the perspective to the H1-norm. This approach benefits from a local collocation technique to estimate the differential operators using the weighted differences over local collection nodes through the RBF expansion. Two test problems illustrate the computational efficiency of the approach. Numerical simulations highlight the performance of the method that provides accurate solutions on complex domains with any distribution node type.

Optimal Control of Robotic Systems Using Finite Elements for Time Integration of Covariant Control Equations

We used an optimal control method involving covariant control equations as optimality conditions, to command the actuators of robot manipulators. These form a coupled system of second order nonlinear ordinary differential equations when associated with the robot motion equations. By solving this system, the control action required to take the robot from an initial to a final state is optimized in a prescribed time. However, the target set of equations exhibited stiffness. Therefore, an adequate solution could only be found for short trajectory durations with readily available numerical methods. We examined a time discretization procedure based on cubic and quintic Hermite finite elements which exhibited superconvergence properties for interpolation. This motivated us to develop a time integration algorithm based on Hermite's technique, where motion and control equations were perturbed to solve the optimal control problem. The optimal motion of a robotic manipulator was simulated using this algorithm. Our method was compared with a commercial differential equations solver on the basis of specific indicators. It outperformed the commercial solver by effectively solving the stiff set of equations for longer trajectory durations, with the cubic elements performing better than the quintic ones in this sense. The convergence analysis of our method confirmed that the quintic elements are more precise at the cost of increased computational burden, but converge at a lower rate than expected. Controlled motion experiments on a robotic manipulator validated our methodology. Trajectories were smoothly tracked and results exposed further methodology improvements.

Rate-independent evolution of sets

<p style='text-indent:20px;'>The goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of (the complement of) a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes. <p style='text-indent:20px;'>In the mathematical modeling of this process, we distinguish the <i>adhesive</i> case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the <i>brittle</i> case, enforcing this constraint. The existence of <i>Energetic solutions</i> for the adhesive system is proved by passing to the limit in the associated time-incremental minimization scheme. In the brittle case, this time-discretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energy-dissipation balance in the time-continuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions.

Building a path-integral calculus: a covariant discretization approach

Path integrals are a central tool when it comes to describing quantum or thermal fluctuations of particles or fields. Their success dates back to Feynman who showed how to use them within the framework of quantum mechanics. Since then, path integrals have pervaded all areas of physics where fluctuation effects, quantum and/or thermal, are of paramount importance. Their appeal is based on the fact that one converts a problem formulated in terms of operators into one of sampling classical paths with a given weight. Path integrals are the mirror image of our conventional Riemann integrals, with functions replacing the real numbers one usually sums over. However, unlike conventional integrals, path integration suffers a serious drawback: in general, one cannot make non-linear changes of variables without committing an error of some sort. Thus, no path-integral based calculus is possible. Here we identify which are the deep mathematical reasons causing this important caveat, and we come up with cures for systems described by one degree of freedom. Our main result is a construction of path integration free of this longstanding problem, through a direct time-discretization procedure.

A time domain finite element approach based on time series estimation

This paper details the use of time series for solving finite element equations. It was aimed at eliminating the discretization of time intervals to reduce calculation time and to obtain accurate results using time-dependent variables. For this purpose, the matrix form of the time series finite element formulation for both continuous and discontinuous time functions were extracted using Taylor and Fourier series for estimating time-dependent functions. Based on the extracted formulation, for solving elastodynamic problems, a computer code and algorithm was introduced. The efficiency and accuracy of the formulation was evaluated by solving different analytical examples. To reveal the power of time series for modeling real problems three practical examples such as settlement, flood protector structures and temporary cofferdams were modeled using time series approach and the results of analyses were presented. The results showed that having removed time discretization procedure, the time series based finite element method can be efficiently applied to decrease the calculation time and computational efforts. In addition, time-continuous responses resulting from this method, help reduce the calculation steps which is vital in many engineering problems.

Analytical Modeling of the Mixed-Mode Growth and Dissolution of Precipitates in a Finite System

In this paper, a novel analytical modeling of the growth and dissolution of precipitates in substitutional alloys is presented. This model uses an existing solution for the shape-preserved growth of ellipsoidal precipitates in the mixed-mode regime, which takes into account the interfacial mobility of the precipitate. The dissolution model is developed by neglecting the transient term in the mass conservation equation, keeping the convective term. It is shown that such an approach yields the so-called reversed-growth approximation. A time discretization procedure is proposed to take into account the evolution of the solute concentration in the matrix as the phase transformation progresses. The model is applied to calculate the evolution of the radius of spherical -Al2Cu precipitates in an Al rich matrix at two different temperatures, for which growth or dissolution occurs. A comparison of the model is made, with the results obtained using the numerical solver DICTRA. The very good agreement obtained for cases where the interfacial mobility is very high indicates that the time discretization procedure is accurate.

A 2D vertical model for simulating surface and subsurface flows using finite element–finite volume methods

AbstractA numerical model is presented for simulation of hydrodynamics of a 2D vertical free surface domain consisting of an arbitrary partitioned porous and non-porous area. To this end, modified Navier–Stokes equations are considered which could be applied in surface water and in subsurface flows, simultaneously. A wide range of Reynolds number has been considered, from which non-Darcy effects have also been taken into account. A fractional step method has been used in the time discretization procedure, where the convection and diffusion terms are separated from the pressure term while solving the momentum equations. To include the variation of surface elevation in computation, the domain has been divided into two parts, namely, ‘interior subdomain’, which never gets dry during the simulation period, covered by fixed unstructured triangular grids and ‘top layer’ with only a one layer structured grid, the position of which varies with the water surface. The validation of the proposed model has been achieved by comparison of its results with both theoretical and experimental data reported in the literature.

Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics

We study the convergence of an alternate minimization scheme for a Ginzburg–Landau phase-field model of fracture. This algorithm is characterized by the lack of irreversibility constraints in the minimization of the phase-field variable; the advantage of this choice, from a computational stand point, is in the efficiency of the numerical implementation. Irreversibility is then recovered a posteriori by a simple pointwise truncation. We exploit a time discretization procedure, with either a one-step or a multi (or infinite)-step alternate minimization algorithm. We prove that the time-discrete solutions converge to a unilateral L2-gradient flow with respect to the phase-field variable, satisfying equilibrium of forces and energy identity. Convergence is proved in the continuous (Sobolev space) setting and in a discrete (finite element) setting, with any stopping criterion for the alternate minimization scheme. Numerical results show that the multi-step scheme is both more accurate and faster. It provides indeed good simulations for a large range of time increments, while the one-step scheme gives comparable results only for very small time increments.

Consistent finite-dimensional approximation of phase-field models of fracture

In this paper we focus on the finite-dimensional approximation of quasi-static evolutions of critical points of the phase-field model of brittle fracture. In a space discretized setting, we first discuss an alternating minimization scheme which, together with the usual time-discretization procedure, allows us to construct such finite-dimensional evolutions. Then, passing to the limit as the space discretization becomes finer and finer, we prove that any limit of a sequence of finite-dimensional evolutions is itself a quasi-static evolution of the phase-field model of fracture. In particular, our proof shows for the first time the consistency of numerical schemes related to the study of fracture mechanics and image processing.

The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations

We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can keep avoiding the local characteristic decompositions for higher derivative terms in Taylor expansion, even omit partly procedure of the nonlinear weights. Extensive simulations are performed, which show that the fifth order finite volume WENO (Weighted Essentially Non-oscillatory) schemes based on Lax–Wendroff-type time discretization provide a higher accuracy order, non-oscillatory properties and more cost efficiency than WENO scheme based on Runge–Kutta time discretization for certain problems. Those conclusions almost agree with that of finite difference WENO schemes based on Lax–Wendroff time discretization for Euler system, while finite volume scheme has more flexible mesh structure, especially for unstructured meshes.

Generalized Primitive Stamps for Nonlinear Circuit-Field Coupling in the Transient Case

The simultaneous solution of circuit and field equations is often required in the analysis of magnetic devices. Although schemes for the solution of this coupled problem have been proposed, the existing formulations are usually tied to specific time discretization or nonlinear iteration expressions and therefore lack generality. In this paper, a highly systematized approach is proposed for strong circuit-field coupling in a transient finite element context, by identifying primitive stamps for field elements, circuit elements, and circuit-field couplings. The filamentary and solid conductors of the field model are the key elements for systematic interconnection using generalized stamps and modified nodal analysis. As a result, the coupling equations are treated just as any other circuit element, simplifying the implementation and providing a unified framework for circuit-field analysis. Both circuit and field stamps are independent of the time discretization and nonlinear solving procedure. With the proposed scheme, the finite element equations can be linked to circuits of arbitrary topology. The theory is developed for the axisymmetric and Cartesian 2-D cases and several examples are then solved to show the effectiveness of our new approach. Solutions are compared with those produced by well-known and validated commercial software packages which implement different, proprietary, and/or undisclosed circuit-field coupling methods.

Dynamic analysis of a thin-walled beam with open cross section subjected to dynamic loads using a high-order implicit algorithm

In this paper, the forced nonlinear dynamic behavior of thin-walled beams with open cross section under external dynamic loads is analyzed by means of a high order implicit algorithm. This algorithm is developed using a 3D nonlinear model that takes into account the large torsion without any assumption on the torsion angle amplitude neither in the constitutive law nor in the derivation for governing dynamic equations. This algorithm is built by employing the following four steps: 1 – the space and time discretization procedures, 2 – a change of variable, 3 – a homotopy transformation, 4 – techniques used in the Asymptotic Numerical Method (ANM) (Cochelin et al., 2007; Mottaqui et al., 2010) [1,2]. The originality of this work reside in the fact that we use, for the first time, this algorithm for nonlinear analysis of thin-walled beams with open cross section under an arbitrary load. The space and time discretizations are performed respectively by the finite elements method and by the classical implicit Newmark scheme. The performance of the high order implicit algorithm is tested on four examples of nonlinear dynamic: a mono-symmetrical beam with a T cross section under external dynamic load, a mono-symmetrical beam with U cross-section under external dynamic load, a bi-symmetrical clamped-free beam IPE300 under harmonic loads and a bi-symmetrical simply supported beam with cruciform section under harmonic loads. A comparison of the obtained results with those computed by the Abaqus industrial code is given. This comparison confirms the robustness, accuracy and efficiency of this high order implicit algorithm.

Time discretization of the point kinetic equations using matrix exponential method and First-Order Hold

A system of nonlinear differential equations is derived to model the dynamics of neutron density and the delayed neutron precursors within a point kinetics equation modeling framework for a nuclear reactor. The point kinetic equations are mathematically characterized as stiff, occasionally nonlinear, ordinary differential equations, posing significant challenges when numerical solutions are sought and traditionally resulting in the need for smaller time step intervals within various computational schemes. In light of the above realization, the present paper proposes a new discretization method inspired by system-theoretic notions and technically based on a combination of the matrix exponential method (MEM) and the First-Order Hold (FOH) assumption. Under the proposed time discretization structure, the sampled-data representation of the nonlinear point kinetic system of equations is derived. The performance of the proposed time discretization procedure is evaluated using several case studies with sinusoidal reactivity profiles and multiple input examples (reactivity and neutron source function). It is shown, that by applying the proposed method under a First-Order Hold for the neutron density and the precursor concentrations at each time step interval, the stiffness problem associated with the point kinetic equations can be adequately addressed and resolved. Finally, as evidenced by the aforementioned detailed simulation studies, the proposed method retains its validity and accuracy for a wide range of reactor operating conditions, including large sampling periods dictated by physical and/or technical limitations associated with the current state of sensor and digital reactor control system technology.

A second order explicit finite difference method for the fractional advection diffusion equation

We develop a numerical method for fractional advection diffusion problems with source terms in domains with homogeneous boundary conditions. The numerical method is derived by using a Lax–Wendroff-type time discretization procedure, it is explicit and second order accurate. The convergence of the numerical method is studied and numerical results are presented.

QUASISTATIC ISOTHERMAL EVOLUTION OF SHAPE MEMORY ALLOYS

This paper focuses on a three-dimensional phenomenological model for the isothermal evolution of a polycrystalline shape memory alloy. The model, originally proposed by Auricchio, Taylor, and Lubliner in 1997, is thermodynamically consistent and reproduces the crucial martensitic reorientation effect as well as the tension-compression asymmetric behavior of the material. We prove the existence of a weak solution of the corresponding quasistatic evolution problem by passing to the limit within a time-discretization procedure.

Simulations of Shallow Water Equations with Finite Difference Lax-Wendroff Weighted Essentially Non-oscillatory Schemes

In this paper we study a Lax-Wendroff-type time discretization procedure for the finite difference weighted essentially non-oscillatory (WENO) schemes to solve one-dimensional and two-dimensional shallow water equations with source terms. In order to maintain genuinely high order accuracy and suit to problems with a rapidly varying bottom topography we use WENO reconstruction not only to the flux but also to the source terms of algebraical modified shallow water equations. Extensive simulations are performed, as a result, the WENO schemes with Lax-Wendroff-type time discretization can maintain nonoscillatory properties and more cost effective than that with Runge-Kutta time discretization.

Efficient preconditioning techniques for finite‐element quadratic discretization arising from linearized incompressible Navier–Stokes equations

AbstractWe develop an efficient preconditioning techniques for the solution of large linearized stationary and non‐stationary incompressible Navier–Stokes equations. These equations are linearized by the Picard and Newton methods, and linear extrapolation schemes in the non‐stationary case. The time discretization procedure uses the Gear scheme and the second‐order Taylor–Hood element P2−P1 is used for the approximation of the velocity and the pressure. Our purpose is to develop an efficient preconditioner for saddle point systems. Our tools are the addition of stabilization (penalization) term r∇(div(·)), and the use of triangular block matrix as global preconditioner. This preconditioner involves the solution of two subsystems associated, respectively, with the velocity and the pressure and have to be solved efficiently. Furthermore, we use the P1−P2 hierarchical preconditioner recently proposed by the authors, for the block matrix associated with the velocity and an additive approach for the Schur complement approximation. Finally, several numerical examples illustrating the good performance of the preconditioning techniques are presented. Copyright © 2009 John Wiley &amp; Sons, Ltd.