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Abstract We study a mechanical system with a finite number of degrees of freedom, subjected to perfect time-dependent frictionless unilateral (possibly nonconvex) constraints with inelastic collisions on active constraints. The dynamic is described in the form of a second-order measure differential inclusion. Under some regularity assumptions on the data, we establish several properties of the set of admissible positions, which is not necessarily convex but assumed to be uniformly prox-regular. Our approach does not require any second-order information or boundedness of the Hessians of the constraints involved in the problem and are specific to moving sets represented by inequalities constraints. On that basis, we are able to discretize our problem by the time-stepping algorithm and construct a sequence of approximate solutions. It is shown that this sequence possesses a subsequence converging to a solution of the initial problem. This methodology is not only used to prove an existence result but could be also used to solve numerically the vibroimpact problem with time-dependent nonconvex constraints.

In this paper, we propose and analyze a second order efficient numerical scheme for the time fractional L 2 gradient flows. The proposed scheme is based on scalar auxiliary variable (SAV) approach and L2- 1 σ time stepping with the so-called sum-of-exponentials (SOE) technique. The main idea is to split the time fractional derivative into two parts: the local part and the history part. Then, we use the L2- 1 σ formula to discretize the local part and adopt the extended SAV approach to deal with the history and nonlinear terms. The unconditional stability of the numerical scheme is rigorously proved for the uniform mesh. The main novelty of the paper is: this is the first proof of the unconditional stability of the L2- 1 σ time stepping schemes with SOE technique for time fractional L 2 gradient flows. Several numerical examples are provided to verify the accuracy and efficiency of the proposed scheme. Finally, the new scheme is applied to investigate the coarsening dynamics governed by the time fractional L 2 gradient flows. [ABSTRACT FROM AUTHOR]

In this article, the existing nonsmooth generalized‐α method for the simulation of mechanical systems with frictionless contacts, modeled as unilateral constraints, is extended to systems with frictional contacts. On that account, we complement the unilateral constraints with set‐valued Coulomb‐type friction laws. Moreover, we devise a set of benchmark systems, which can be used to validate numerical schemes for mechanical systems with frictional contacts. Finally, this set of benchmarks is used to numerically assert the properties striven for during the derivation of the presented scheme. Specifically, we show that the presented scheme can reproduce the dynamics of the frictional contact adequately and no numerical penetration of the contacting bodies arises—a big issue for most popular time‐stepping schemes such as the one of Moreau. Moreover, we demonstrate that the presented scheme performs well for multibody systems containing flexible parts and that it allows general parametrizations such as the use of unit quaternions for the rotation of rigid bodies. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze high-order efficient schemes for the time-fractional Allen-Cahn equation. The proposed schemes are based on the L1 discretization for the time-fractional derivative and the extended scalar auxiliary variable (SAV) approach developed very recently to deal with the nonlinear terms in the equation. The main contributions of the paper consist of (1) constructing first- and higher order unconditionally stable schemes for different mesh types, and proving the unconditional stability of the constructed schemes for the uniform mesh; (2) carrying out numerical experiments to verify the efficiency of the schemes and to investigate the coarsening dynamics governed by the time-fractional Allen-Cahn equation. In particular, the influence of the fractional order on the coarsening behavior is carefully examined. Our numerical evidence shows that the proposed schemes are more robust than the existing methods, and their efficiency is less restricted to particular forms of the nonlinear potentials. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a new class of schemes based on a variant of the scalar auxiliary variable (SAV) approaches for gradient flows. Precisely, we construct more robust first and second order unconditionally stable schemes by introducing a new defined auxiliary variable to deal with nonlinear terms in gradient flows. The new approach consists in splitting the gradient flow into decoupled linear systems with constant coefficients, which can be solved using existing fast solvers for the Poisson equation. This approach can be regarded as an extension of the SAV method; see, e.g., Shen et al. (2018) [21] , in the sense that the new approach comes to be the conventional SAV method when α = 0 and removes the boundedness assumption on ∫ Ω F (ϕ) d x required by the SAV. The new approach only requires that the total free energy or a part of it is bounded from below, which is more realistic in physically meaningful models. The unconditional stability is established, showing that the efficiency of the new approach is less restricted to particular forms of the nonlinear terms. A series of numerical experiments is carried out to verify the theoretical claims and illustrate the efficiency of our method. [ABSTRACT FROM AUTHOR]

In the beginning of the 80s, a rigorous mathematical framework was developed for the dynamics of multibody systems with perfect unilateral contacts, particularly due to the contributions of Schatzman and Moreau. Efficient numerical methods have been proposed, for instance Moreau’s NonSmooth Contact Dynamics (NSCD) (Moreau, 1999), which was then extended by Jean to cases with friction (Jean, 1999). But the algorithm, in the latter case, is no longer the time discretization of an evolution problem. In this work, we derive a new algorithm from the time discretization of an evolution problem for multibody dynamics with contacts and friction. Our algorithm has many points in common with the one of Jean and Moreau, but it converges reliably and fixes some energetic inconsistencies. The similarities and differences between the algorithms are illustrated on three planar archetypal examples. [ABSTRACT FROM AUTHOR]

In computational structural dynamics, particularly in the presence of nonsmooth behavior, the choice of the time-step and the time integrator has a critical impact on the feasibility of the simulation. Furthermore, in some cases, as in the case of a bridge crane under seismic loading, multiple time-scales coexist in the same problem. In that case, the use of multi-time scale methods is suitable. Here, we propose a new explicit-implicit heterogeneous asynchronous time integrator (HATI) for nonsmooth transient dynamics with frictionless unilateral contacts and impacts. Furthermore, we present a new explicit time integrator for contact/impact problems where the contact constraints are enforced using a Lagrange multiplier method. In other words, the aim of this paper consists in using an explicit time integrator with a fine time scale in the contact area for reproducing high frequency phenomena, while an implicit time integrator is adopted in the other parts in order to reproduce much low frequency phenomena and to optimize the CPU time. In a first step, the explicit time integrator is tested on a one-dimensional example and compared to Moreau-Jean's event-capturing schemes. The explicit algorithm is found to be very accurate and the scheme has generally a higher order of convergence than Moreau-Jean's schemes and provides also an excellent energy behavior. Then, the two time scales explicit-implicit HATI is applied to the numerical example of a bridge crane under seismic loading. The results are validated in comparison to a fine scale full explicit computation. The energy dissipated in the implicit-explicit interface is well controlled and the computational time is lower than a full-explicit simulation. [ABSTRACT FROM AUTHOR]

• A state-space representation of the Xinanjiang model is derived. • The runoff generation process-related submodules amplify numerical errors. • The runoff concentration process-related submodules could alleviate numerical errors. • A large numerical error occurs with intensive precipitation and decreases afterward. The numerical error of the Xinanjiang (XAJ) model has long been ignored, despite its wide application in China. Separating the mathematically formulated model from the computationally formulated model and further solving by an appropriate time-stepping scheme provides a chance to systematically examine the numerical error of the XAJ model because the exact analytical solution is difficult to obtain owing to its nonlinearity. The mathematically formulated model is proposed by deriving a state-space representation of the XAJ model by identifying the state variables, fluxes, and governing differential equations. Five experiments were conducted around the original XAJ model and its state-space representation with different parameter groups, input series, and time-stepping schemes to examine the numerical error at the submodule and model levels. We found that only the tension water storage capacity curve submodule of the original XAJ model is numerical error-free, and the adaptive-step explicit fourth-order Runge-Kutta scheme is the most efficient because it balances the accuracy, efficiency, and physical constraint preservation. The numerical error of the total amount of the discharge at the watershed outlet within a time period (Q ∗) produced by the original XAJ model is relatively large, which mainly comes from the free water storage capacity curve submodule. The large numerical error of Q ∗ occurs with high precipitation intensity, and decreases as the flood receded. The submodules related to the runoff generation process introduce and amplify numerical errors, while the submodules related to the runoff concentration process has the ability to alleviate numerical errors. This study helps to deepen our understanding of the numerical error of the XAJ model, and thus guides its application and modification. [ABSTRACT FROM AUTHOR]

We analyze an h- p version Petrov-Galerkin finite element method for linear Volterra integrodifferential equations. We prove optimal a priori error bounds in the L- and H-norm that are explicit in the time steps, the approximation orders and in the regularity of the exact solution. Numerical experiments confirm the theoretical results. Moreover, we observe that the numerical scheme superconverges at the nodal points of the time partition. [ABSTRACT FROM AUTHOR]

We consider a system of rigid bodies subjected to some non penetration conditions characterized by the inequalities ƒα(q) ≥ 0, α ∈ {1,...,v}, v ≥ 1, for the configuration q ∈ ℝd. We assume that there is no adhesion and R no friction during contact and we model the behaviour of the system at impact by a Newton's law. Starting from the mechanical description of the problem, we derive two mathematical formulations, using either the configuration or the generalized velocity as unknown. Then a velocity-based time-stepping scheme, inspired by the catching-up algorithms, is presented and its convergence in the multi-constraint case (i.e v ≥ 1) is stated. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce a method for the numerical solutions of initial value problems, that combines finite differences with Simpson’s rule. The effectiveness of the method is proved by solving, in one spatial dimension, a stiff and convection-dominated transport problem. To solve the same problem in two spatial dimensions, the proposed method was used successfully in combination with Strang’s operator decomposition method. [ABSTRACT FROM AUTHOR]

This paper describes a flexible time-stepping scheme for a recently developed hybrid field-circuit solver based on the extended time-domain finite element method (TDFEM) to alleviate the limitation on the use of a system-wide global time-step size. The proposed time-stepping scheme generalizes the strict synchronous coupling mechanism between the FEM and circuit subsystems and allows the signals in the different subsystems to be tracked and sampled at different time-step sizes. The signals from a slow subsystem with a larger time-step size are extrapolated, when necessary, for updating the signals in a fast subsystem with a smaller time-step size. The capability of the hybrid field-circuit solver with the proposed time-stepping scheme is further enhanced by the application of a tree-cotree splitting technique to the FEM subsystem, which helps reduce the iteration count per time step for a preconditioned iterative solution when the time-step size of the FEM subsystem becomes relatively large. With the flexibility of choosing subsystem-specific time-step sizes, the proposed time-stepping scheme improves the computational efficiency of the existing TDFEM-based hybrid field-circuit solver especially when the computational cost associated with the slow subsystems is much higher than that associated with the fast subsystems. [ABSTRACT FROM AUTHOR]

Abstract: The dynamics of multibody systems with friction-affected bilateral constraints is essentially different from those of smooth multibody systems. In this paper, general modeling and numerical methods for this kind of friction-affected system are given. Each friction-affected bilateral constraint is modeled by splitting it into two unilateral constraints opposite to each other. The constraint equalities are replaced with complementarity inequalities to avoid the absolute value terms in the dynamic equations. A linear complementarity problem time-stepping scheme is presented for simulation that does not suffer from the difficulties of enumeration (known as Delassus’ problem). The algorithm has the advantage that it needs no detection for stick-slip transition when neglecting the difference between the static and kinetic friction coefficients. Our method is carried out in an example to analyze the nonsmooth nonlinear behavior of a typical mechanism. [Copyright &y& Elsevier]

We analyze a coordinate-free tensor setting in ℝ3 within the context of the classical tensor analysis. To this end, we formulate in a basis-free manner the notions of second- and fourth-rank tensors in ℝ3, and corresponding operations on tensors. Among the large number of different approaches to the tensor setting, we give the preference to the convenient ones, concerning the specific needs of computational solid mechanics. We use the well-known Fréchet derivative to define the derivative of a function with respect to its tensor argument in a natural way. Furthermore, such aspects as the derivative with respect to a symmetric tensor argument and its uniqueness are covered in this paper. For the sake of completeness we present the coordinate representation of tensors and tensor operations. This representation is obtained in a straight-forward manner from the coordinate-free one. In particular, we elaborate the computation of the inverse of a fourth-rank tensor and the inverse of a linear transformation on the space of symmetric second-rank tensors. The tensor formalism is applied to the analysis of a nonlinear system of differential and algebraic equations governing visoplastic material response. An implicit time-stepping algorithm is formulated and the numerical treatment of the algorithm is discussed. [ABSTRACT FROM AUTHOR]

Based on Green’s theorem, a time domain numerical model was constructed to simulate wave making phenomenon caused by a moving ship. In this article, the Rankine sources and dipoles were placed on boundary surfaces (i.e., the ship surface and free surface), and a time-stepping scheme was employed. Its unique characteristic is that steady state can be realized from initial value by employing the time-stepping scheme and unsteady free surface conditions. In time domain, if the results of unsteady flow problem tend to data stabilization after many time steps of computation, they could be regarded as the data of steady ones. This model could be employed to steady or unsteady problems. Theoretical reasoning and computational process of this method was described in detail. The linear and nonlinear boundary conditions on body surface were studied, and the relative means to realize these boundary conditions in iterative computation were also discussed. Some proper parameters about the model of the Wigley hull were determined by many numerical tests, and their influences on wave making resistance and wave pattern were discussed. According to the comparison between numerical results and data available in relative references, the method used in this work is proven to be a reliable method in time domain. And the lattice reorganization in every time step computation is a feasible numerical approach. [ABSTRACT FROM AUTHOR]

We calculate numerically the motion of a slender bar dropped on a rigid foundation. For the computation the bar is discretized by a system of rigid bodies linked by spiral springs or by a pair of linear springs. We assume that the impact is frictionless and we model it by Newton’s law. We compute the motion by using either an event-driven method based on the detection of impacts or a time-stepping scheme avoiding the detection of impacts. We calculate also the apparent restitution coefficient and we compare our results with the experimental and numerical results of Stoianovici and Hurmuzlu. [Copyright &y& Elsevier]

In this work a large-eddy simulation (LES) of a turbulent hydrogen jet diffusion flame is presented. The numerical method handles fluctuations of density in space and in time, but assumes density to be independent of pressure (incompressibility). The chemical composition of the fluid is described by solving the filtered transport equation for mixture fraction f. Density, viscosity and temperature are evaluated assuming chemical equilibrium. To account for sub-grid fluctuations of f, its sub-grid distribution is presumed to have the shape of a β-function. The results of the simulation are discussed extensively. The influence of inlet boundary conditions is addressed and radial profiles at different axial positions are shown for a complete set of one-point statistical data. Agreement of numerical results and experimental data is very good. Furthermore, a comparison of Reynolds- and Favre-averages is done and energy spectra at different locations in the flame are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the time fractional molecular beam epitaxial growth models and its numerical solutions. Basically, two types of numerical schemes are proposed and analyzed: one is based on the Crank–Nicolson formula while the other is a kind of backward differentiation formula. The former is proved to be unconditional energy stable with a modified energy on the general temporal mesh, including the graded mesh and the uniform mesh. However, A drawback is its sensitivity to the coefficient and the time step size. The other, i.e., the backward differentiation based scheme, is more robust in term of this sensitivity. Its unconditional stability with respect to the modified energy is proved only for the uniform mesh. The key of these results is the use of a new approach to the time fractional derivative. The idea is to first split the time fractional derivative into the local part and the history part, then treat the two parts in different ways. This splitting allows to derive a nonlocal free energy dispassion for the continuous solution, and makes the construction of stable numerical schemes much easier. It is found that the well known L1, L1–CN, L 1 + –CN and L2 formula with the sum-of-exponentials approach for the local derivative term, combined with the extended auxiliary approach for the nonlinear and history terms result in unconditional stable schemes, which satisfy the modified energy dispassion at the discrete level. Finally, a series of numerical examples are provided to verify the efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]