A C 0 conforming dg finite element method for biharmonic equations on triangle/tetrahedron

A C 0 conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the biharmonic equation. The first strong gradient of C 0 finite element functions is a vector of discontinuous piecewise polynomials. The second gradient is the weak gradient of discontinuous piecewise polynomials. This method, by its name, uses nonconforming (non C 1) approximations and keeps simple formulation of conforming finite element methods without any stabilizers. Optimal order error estimates in both a discrete H 2 norm and the L 2 norm are established for the corresponding finite element solutions. Numerical results are presented to confirm the theory of convergence.

Time-Varying Formation Control for Heterogeneous Planar Underactuated Multi-Vehicle Systems

This paper presents a distributed control approach for time-varying formation of heterogeneous planar underactuated vehicle networks without global position measurements. All vehicles in the network are modeled as generic three degree of freedom planar rigid bodies with two control inputs, and are allowed to have non-identical dynamics. Feasible trajectories are generated for each vehicle using the nonholonomic constraints of the vehicle dynamics. By exploiting the cascaded structure of the planar vehicle model, a transformation is introduced to define the reduced order error dynamics, and then, a sliding-mode control law is proposed. Low level controller for each vehicle is derived such that it only requires relative position and local motion information of its neighbors in a given directed communication network. The proposed formation control law guarantees the uniform global asymptotic stability (UGAS) of the closed-loop system subject to bounded uncertainties and disturbances. The proposed approach can be applied to underactuated vehicle networks consisting of mobile robots, surface vessels and planar aircraft. Simulations are presented to demonstrate the effectiveness of the proposed control scheme.

Numerical analysis of doubly-history dependent variational inequalities in contact mechanics

This paper is devoted to numerical analysis of doubly-history dependent variational inequalities in contact mechanics. A fully discrete method is introduced for the variational inequalities, in which the doubly-history dependent operator is approximated by repeated left endpoint rule and the spatial variable is approximated by the linear element method. An optimal order error estimate is derived under appropriate solution regularities, and numerical examples illustrate the convergence orders of the method.

A parabolic local problem with exponential decay of the resonance error for numerical homogenization

This paper aims at an accurate and efficient computation of effective quantities, e.g. the homogenized coefficients for approximating the solutions to partial differential equations with oscillatory coefficients. Typical multiscale methods are based on a micro–macro-coupling, where the macromodel describes the coarse scale behavior, and the micromodel is solved only locally to upscale the effective quantities, which are missing in the macromodel. The fact that the microproblems are solved over small domains within the entire macroscopic domain, implies imposing artificial boundary conditions on the boundary of the microscopic domains. A naive treatment of these artificial boundary conditions leads to a first-order error in [Formula: see text], where [Formula: see text] represents the characteristic length of the small scale oscillations and [Formula: see text] is the size of microdomain. This error dominates all other errors originating from the discretization of the macro and the microproblems, and its reduction is a main issue in today’s engineering multiscale computations. The objective of this work is to analyze a parabolic approach, first announced in A. Abdulle, D. Arjmand, E. Paganoni, C. R. Acad. Sci. Paris, Ser. I, 2019, for computing the homogenized coefficients with arbitrarily high convergence rates in [Formula: see text]. The analysis covers the setting of periodic microstructure, and numerical simulations are provided to verify the theoretical findings for more general settings, e.g. non-periodic microstructures.

Wave- and drag-driven subharmonic responses of a floating wind turbine

The nonlinear hydrodynamic responses of a novel spar-type soft-moored floating offshore wind turbine are investigated via analysis of motion measurements from a wave-basin campaign. A prototype of the TetraSpar floater, supporting a $1:60$ scale model of the DTU 10 MW reference wind turbine, was subjected to irregular wave forcing (with no wind) and shown to exhibit subharmonic resonant motions, which greatly exceeded the wave-frequency motions. These slow-drift responses are excited nonlinearly, since the rigid-body natural frequencies of the system lie below the incident-wave frequency range. Pitch motion is examined in detail, allowing for identification of different hydrodynamic forcing mechanisms. The resonant response is found to contain odd-harmonic components, in addition to the even harmonics expected a priori and excited by second-order difference-frequency hydrodynamic interactions. Data analysis utilising harmonic separation and signal conditioning suggests that Morison drag excitation or third-order subharmonic potential flow forcing could be at play. In the extreme survival-conditions sea state, the odd resonant responses are identified to be drag-driven. Their importance for the tested floater is appreciable, as their magnitude is comparable to the second-order potential flow effects. Under such severe conditions, the turbine would not be operating, and as such neglecting aerodynamic forcing and motion damping is likely to be reasonable. Additionally, other possible drivers of the resonant pitch response are explored. Both Mathieu-type parametric excitation and wavemaker-driven second-order error waves are found to have negligible influence. However, we note slight contamination of the measurements arising from wave-basin sloshing.

A bias generating temporal distortions in serial perception

A precise estimation of event timing is essential for survival. Yet, temporal distortions are ubiquitous in our daily sensory experience. A specific type of temporal distortion is the time order error (TOE), which occurs when estimating the duration of events organized in a series. TOEs shrink or dilate objective event duration. Understanding the mechanics of subjective time distortions is fundamental since we perceive events in a series, not in isolation. In previous work, we showed that TOEs appear when discriminating small duration differences (20 or 60 ms) between two short events (Standard, S and Comparison, C), but only if the interval between events is shorter than 1 second. TOEs have been variously attributed to sensory desensitization, reduced temporal attention, poor sensory weighting of C relative to S, or idiosyncratic response bias. Surprisingly, the serial dynamics of relative event duration were never considered as a factor generating TOEs. In two experiments we tested them by swapping the order of presentation of S and C. Bayesian hierarchical modelling showed that TOEs emerge when the first event in a series is shorter than the second event, independently of event type (S or C), sensory precision or individual response bias. Participants disproportionately expanded first-position shorter events. Significantly fewer errors were made when the first event was objectively longer, confirming the inference of a strong bias in perceiving ordered event durations. Our finding identifies a hitherto unknown duration-dependent encoding inefficiency in human serial perception.

ON COMPACT 4TH ORDER FINITE-DIFFERENCE SCHEMES FOR THE WAVE EQUATION

We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the n-dimensional nonhomogeneous wave equation, n≥ 1. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for n≥ 2. The alternative technique is applicable to other types of PDEs including parabolic and time-dependent Schro¨dinger ones. The schemes are implicit and three-point in each spatial direction and time and include a scheme with a splitting operator for n≥ 2. For n = 1 and the mesh on characteristics, the 4th order scheme becomes explicit and close to an exact four-point scheme. We present a conditional stability theorem covering the cases of stability in strong and weak energy norms with respect to both initial functions and free term in the equation. Its corollary ensures the 4th order error bound in the case of smooth solutions to the IBVP. The main schemes are generalized for non-uniform rectangular meshes. We also give results of numerical experiments showing the sensitive dependence of the error orders in three norms on the weak smoothness order of the initial functions and free term and essential advantages over the 2nd approximation order schemes in the non-smooth case as well.

Numerical discretization and fast approximation of a variably distributed-order fractional wave equation

We investigate a variably distributed-order time-fractional wave partial differential equation, which could accurately model, e.g., the viscoelastic behavior in vibrations in complex surroundings with uncertainties or strong heterogeneity in the data. A standard composite rectangle formula of mesh size $\sigma$ is firstly used to discretize the variably distributed-order integral and then the L-1 formula of degree of freedom $N$ is applied for the resulting fractional derivatives. Optimal error estimates of the corresponding fully-discrete finite element method are proved based only on the smoothness assumptions of the data. To maintain the accuracy, setting $\sigma=N^{-1}$ leads to $O(N^3)$ operations of evaluating the temporal discretization coefficients and $O(N^2)$ memory. To improve the computational efficiency, we develop a novel time-stepping scheme by expanding the fractional kernel at a fixed fractional order to decouple the fractional operator from the variably distributed-order integral. Only $O(\log N)$ terms are needed for the expansion without loss of accuracy, which consequently reduce the computational cost of coefficients from $O(N^3)$ to $O(N^2\log N)$ and the corresponding memory from $O(N^2)$ to $O(N\log N)$. Optimal-order error estimates of this time-stepping scheme are rigorously proved via novel and different techniques from the standard analysis procedure of the L-1 methods. Numerical experiments are presented to substantiate the theoretical results.

DERIVATION AND IMPLEMENTATION OF A THREE-STEP HYBRID BLOCK ALGORITHM (TSHBA) FOR DIRECT SOLUTION OF LINEAR AND NONLINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

The development and application of an implicit hybrid block method for the direct solution of second order ordinary differential equations with given initial conditions is shown in this research. The derivation of the three-step scheme was done through collocation and interpolation of power series approximation to give a continuous linear multistep method. The evaluation of the continuous method at the grid and off grid points formed the discrete block method. The basic properties of the method such as order, error constant, zero stability, consistency and convergence were properly examined. The new block method produced more accurate results when compared with similar works carried out by existing authors on the solution of linear and non-linear second order ordinary differential equations