Norm Solution Research Articles

Numerical approach for partial eigenstructure assignment problems in singular vibrating structure using active control

In the paper, the partial eigenstructure assignment problems are investigated using acceleration–velocity–displacement active control in a singular vibrating structure. The problems are transformed into solving matrix equations using the receptance matrix method. Iterative sequences are constructed, and the iterative feasibility is presented for solving the matrix equations. The partial eigenvectors of the closed-loop system are reassigned by imposing modal constraints. An algorithm is proposed to get numerical solutions of the derived matrix equations. The initial value condition is discussed to obtain the minimum norm solution of the partial eigenstructure assignment problems. The designed acceleration–velocity–displacement active control can solve the partial eigenstructure assignment problems depending only on original vibrating structure information. The proposed numerical algorithm can obtain the minimum norms of controller gain, which implies minimum energy consumption. Numerical examples are given to illustrate the effectiveness of the proposed methods.

Superconvergence analysis of Galerkin method for semilinear parabolic integro-differential equation

In this paper, based on the bilinear element used for spatial discretization and a linearized backward Euler scheme used for temporal discretization, the superconvergence error estimate is derived for semilinear parabolic integro-differential equation without certain time-step restrictions. The key is to derive a uniform boundness of the numerical solution in energy norm under the weaker assumption compared to previous literatures for nonlinear term. Numerical results are presented to confirm the correctness of the theoretical analysis.

COVID-19 and the global acceleration of digital psychiatry

COVID-19 and the global acceleration of digital psychiatry

Backward Euler method for the equations of motion arising in Oldroyd model of order one with nonsmooth initial data

Abstract In this paper, a backward Euler method combined with finite element discretization in spatial direction is discussed for the equations of motion arising in the two-dimensional Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in $L^{\infty }$ in time. It is shown that the estimates of the discrete solution in Dirichlet norm is bounded uniformly in time. Optimal a priori error estimate in $\textbf {L}^2$-norm is derived for the discrete problem with nonsmooth initial data. This estimate is shown to be uniform in time, under the assumption of uniqueness condition. Finally, we present some numerical results to validate our theoretical results.

Global well-posedness of a Navier–Stokes–Cahn–Hilliard system with chemotaxis and singular potential in 2D

We study a diffuse interface model that describes the dynamics of incompressible two-phase flows with chemotaxis effects. This model also takes into account some significant mechanisms such as active transport and nonlocal interactions of Oono's type. The system under investigation couples the Navier–Stokes equations for the fluid velocity, a convective Cahn–Hilliard equation with physically relevant singular potential for the phase-field variable and an advection-diffusion-reaction equation for the nutrient density. For the initial boundary value problem in a smooth bounded domain Ω ⊂ R 2 , we first prove the existence and uniqueness of global strong solutions that are strictly separated from the pure states ±1 over time. Then we prove the continuous dependence with respect to the initial data and source term for the strong solution in energy norms. Finally, we show the propagation of regularity for global weak solutions.

Optimal l∞ error estimates of the conservative scheme for two-dimensional Schrödinger equations with wave operator

In this work, we consider the numerical computation for the two-dimensional generalized nonlinear Schrödinger equations with wave operator. Based on the scalar auxiliary variable (SAV) approach, the original problem is transformed into an equivalent one, which corresponds to the energy-conservation laws. We present an energy-conserving and linearly implicit three-level scheme for the equivalent system. The energy-conserving property, boundedness of the numerical solution and convergence analysis in the discrete maximum norm are derived, which has not restricted to specific forms of cubic nonlinear term f and not needed the sharply restriction on mesh size. Finally, numerical experiments on several models confirm our theoretical results.

Minimum norm partial eigenstructure assignment problems in high‐order system via feedback control

AbstractIn this article, the minimum norm partial eigenstructure assignment problem (MNPESAP) is considered in a symmetric high‐order system using feedback control law with the highest‐order derivative term. Two new orthogonality relations are constructed using the system matrices and undetermined feedback gains. The parametric expressions of the feedback controller and the necessary and sufficient condition for solving the partial eigenstructure assignment problem are derived. The Sylvester matrix equations and gradient formula are presented for solving MNPESAP. An algorithm is proposed to obtain the minimum norm solutions of feedback gains using gradient‐based optimization method. Numerical examples are provided to verify the effectiveness of given results.

An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

This paper presents a nonconforming finite element scheme for the planar biharmonic equation, which applies piecewise cubic polynomials (P3) and possesses \({\cal O}({h^2})\) convergence rate for smooth solutions in the energy norm on general shape-regular triangulations. Both Dirichlet and Navier type boundary value problems are studied. The basis for the scheme is a piecewise cubic polynomial space, which can approximate the H4 functions with \({\cal O}({h^2})\) accuracy in the broken H2 norm. Besides, a discrete strengthened Miranda-Talenti estimate (∇ 2h ·, ∇ 2h ·) = (∆h ·, ∆h ·), which is usually not true for nonconforming finite element spaces, is proved. The finite element space does not correspond to a finite element defined with Ciarlet’s triple; however, it admits a set of locally supported basis functions and can thus be implemented by the usual routine. The notion of the finite element Stokes complex plays an important role in the analysis as well as the construction of the basis functions.

High order discontinuous Galerkin method for reduced flow models in fractured porous media

We construct a new symmetric interior penalty discontinuous Galerkin (SIPDG)-method for incompressible flow in fractured porous media. The method is developed for computing accurate approximations of the reduced flow model, where fractures are treated as lower dimensional objects. Unlike previous formulations, the methodology proposed herein shows ability to capture the asymptotic limits of highly permeable and fracture seals, also covering a wider range of values of the quadrature integration parameter which appears in the pressure jumps across the fractures. As a first novel contribution we should mention that the method was developed for specific interface condition in the reduced problem for which known in literature DG method are not applicable. As a second novelty we propose a new penalty technique for stabilization of the method and obtain explicit estimate for the penalty parameters associated with flow in matrix and fractures in order to achieve stability. And finally we derive new high order hp type a priori error estimates for the numerical solution in the energy norm. Numerical results illustrate the performance of the proposed SIPDG-method in simulating discrete fracture models.

DIMENSIONAL SYNTHESIS OF A GAIT REHABILITATION CABLE-SUSPENDED ROBOT ON MINIMUM 2-NORM TENSIONS

A new design of gait rehabilitation robot with cable-suspended configuration is proposed. Due to the under-constrained nature, it enables reducing the constraint feeling of patients. Cables are attached to cuffs mounted on the leg. A detailed mechanical design is presented and a kinematics model is developed. Dimensional synthesis is performed in two steps. First, the cable disposition should be determined within a range to maintain cable-suspended configuration using the minimum 2-norm solution of tensions. Second, the optimal cable disposition is achieved with the Root Mean Square of tension solutions. Gait rehabilitation robots with three or four cables are discussed and compared to determine dimensional parameters in terms of the locations of pulleys. A simulation model with ADAMS software is presented and the cable module is utilized to imitate the cable-driven system in real. Tension distribution is obtained from the simulation model, which is employed in comparison with the calculated values. The simulation results demonstrate the effectiveness of the presented method.

Compact Difference Schemes on a Three-Point Stencil for Second-Order Hyperbolic Equations

We consider compact difference schemes of approximation order 4+2 on a three-point spatial stencil for theKlein–Gordon equations with constant and variable coefficients. New compact schemes areproposed for one type of second-order quasilinear hyperbolic equations. In the case of constantcoefficients, we prove the strong stability of the difference solution under small perturbations ofthe initial conditions, the right-hand side, and the coefficients of the equation. A priori estimatesare obtained for the stability and convergence of the difference solution in strong mesh norms.

PERIDYNAMIC METHOD BASED ON NONLOCAL LSM OPTIMIZATION AND DEFORMATION SIMULATION OF BRITTLE MATERIALS

Based on the peridynamic model, the relationship between the micro modulus in peridynamics and the elastic constants in classical theory is established by assuming small deformation. The kernel function which can reflect the nonlocal action characteristics is introduced to improve the calculation accuracy, and the linear equations related to the micro modulus are established by means of stiffness equivalence. The least square minimum (LSM) norm solution of uncertain linear equations is used to optimize the micro modulus in peridynamics, and the optimal nonnegative modulus is obtained through quadratic programming. The deformation of two-dimensional plate under uniaxial and biaxial loads and the crack growth of brittle material with prefabricated cracks under loads are simulated by the optimized method, and the results are compared with those by classical peridynamics method. The results show that the optimized method can better reflect the deformation and failure characteristics of the structure under load conditions, and the maximum error and error range of material deformation simulation are better than the classical method; and moreover, the simulation of crack growth process has better convergence speed and convergence results under the same calculation cost, which verifies the effectiveness and the wide application prospect of the proposed method.

An iterative algorithm for generalized periodic multiple coupled Sylvester matrix equations

The paper is indicated to constructing a modified conjugate gradient iterative (MCG) algorithm to solve the generalized periodic multiple coupled Sylvester matrix equations. It can be proved that the proposed approach can find the solution within finite iteration steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the least Frobenius norm solution of the system. Some numerical examples are illustrated to show the performance of the proposed approach and its superiority over the existing method CG.

A Distribution Method of Driving Torque for a Novel 3UPS-RR Redundant Solar Tracker

Abstract This paper proposes a novel two-axis solar tracker with a redundant parallel mechanism and investigates the distribution method of driving torque. In view of the difference between the singular configuration of the redundant parallel mechanisms and that of the corresponding non-redundant ones, an index related to the minimum singular value of the Jacobian matrix is used to indicate the position of the singular configuration relative to the boundaries of the required workspace. The driving torque and energy consumption can be optimized with this index. Based on the fact that the direction of driving torque is opposite to that of rotor in most of the running processes, a distribution method of driving torque with the minimum energy consumption for the redundant parallel solar tracker is proposed. The distribution method is compared with the minimum norm solution which is adopted by the conventional redundant parallel mechanism. And energy consumption can be significantly reduced by adopting this method. In addition, the workspace and energy consumption of the redundant solar tracker and its non-redundant counterpart are compared. The results show that the redundant solar tracker has a larger workspace and lower energy consumption.

Iteratively reweighted least squares and slime mold dynamics: connection and convergence

We present a connection between two dynamical systems arising in entirely different contexts: the Iteratively Reweighted Least Squares (IRLS) algorithm used in compressed sensing and sparse recovery to find a minimum $$\ell _1$$ -norm solution in an affine space, and the dynamics of a slime mold (Physarum polycephalum) that finds the shortest path in a maze. We elucidate this connection by presenting a new dynamical system – Meta-Algorithm – and showing that the IRLS algorithms and the slime mold dynamics can both be obtained by specializing it to disjoint sets of variables. Subsequently, and building on work on slime mold dynamics for finding shortest paths, we prove convergence and obtain complexity bounds for the Meta-Algorithm that can be viewed as a “damped” version of the IRLS algorithm. A consequence of this latter result is a slime mold dynamics to solve the undirected transshipment problem that computes a $$(1+\varepsilon )-$$ approximate solution in time polynomial in the size of the input graph, maximum edge cost, and $$\frac{1}{\varepsilon }$$ – a problem that was left open by the work of (Bonifaci V et al. [10] Physarum can compute shortest paths. Kyoto, Japan, pp. 233–240).

Strong convergence of double-projection method for variational inequality problems

The paper proposes a numerical method for solving a variational inequality problem (VIP) involving a monotone and Lipschitz continuous operator in a Hilbert space. The method can be considered as a combination between an extragradient method and a regularization method. We prove that the iterative sequences generated by our method converge strongly to the smallest norm solution of problem VIP. Obtaining the strong convergence of the iterative sequences is based on regularization solutions when the regularization parameters are chosen suitably. Our numerical experiments are implemented to illustrate the behavior of the new method. The numerical results have shown the effectiveness and fast convergence of the method over existing methods.

Does the upper bound solution property of the Node-based Smoothed Point Interpolation Methods (NSPIMs) hold true in coupled flow-deformation problems of porous media?

It has been shown that a recently developed class of numerical solutions referred to as node-based smoothed point interpolation methods (NSPIMs) can provide an upper bound solution in energy norm for force-driven elasticity problems. In this study, bound solution property of the NSPIMs is examined in plane-strain coupled problems of porous media, to explore how the presence of pore water pressure affects this feature of the NSPIMs. Brief descriptions of the governing differential equations and the formulation of the NSPIMs are first presented, followed by the discretised form of the governing equations. The bound solution properties of three popular NSPIMs in poro-elasticity problems are then thoroughly investigated, both theoretically and numerically, and compared to those of the linear FEM. The results show that the upper bound solution property of the NSPIMs is lost in coupled problems of poro-elasticity due to the coupling effects of the fluid phase.

RoDeRain: Rotational Video Derain via Nonconvex and Nonsmooth Optimization

Video derain is an important issue in the field of digital image processing and computer vision. This paper divides rain streaks into two types: one is rain in natural scenes, and the other is rain in stochastic scenes. In this paper, we propose a novel rotational video derain algorithm via nonconvex and nonsmooth algorithm (RoDerain). Not only can the rain streaks in natural scene be removed, but the rain streaks in stochastic scene can be also well removed. This paper added the rotation operator based on the discriminatively intrinsic priors of rain streaks and clean videos to remove the rain streaks in both natural and stochastic scenes.For the low rank problem of the background, we replace the solution of the nuclear norm with improved IRNN-Capped L1 suitable for tensor. Finally,this paper used the Alternating Direction Method of Multipliers (ADMM) to optimize the solution of the proposed rain streaks removal algorithm model.The disadvantage is that global information is not considered. And the extensive experiment results show that our proposed algorithm performs favorably in comparison to several popular rain removal algorithms.

Decay Estimates for Nonradiative Solutions of the Energy-Critical Focusing Wave Equation

We consider the energy-critical focusing wave equation in space dimension \(N\ge 3\). The equation has a nonzero radial stationary solution W, which is unique up to scaling and sign change. It is conjectured (soliton resolution) that any radial, bounded in the energy norm solution of the equation, behaves asymptotically as a sum of modulated Ws, decoupled by the scaling, and a radiation term. A nonradiative solution of the equation is by definition a solution of which energy in the exterior \(\{|x|>|t|\}\) of the wave cone vanishes asymptotically as \(t\rightarrow +\infty \) and \(t\rightarrow -\infty \). In our previous work [9], we have proved that the only radial nonradiative solutions of the equation in three space dimensions are, up to scaling, 0 and \(\pm W\). This was crucial in the proof of soliton resolution in [9]. In this paper, we prove that the initial data of a radial nonradiative solution in odd space dimension have a prescribed asymptotic behavior as \(r\rightarrow \infty \). We will use this property for the proof of soliton resolution, for radial data, in all odd space dimensions. The proof uses the characterization of nonradiative solutions of the linear wave equation in odd space dimensions obtained by Lawrie, Liu, Schlag, and the second author in [15]. We also study the propagation of the support of nonzero radial solutions with compactly supported initial data and prove that these solutions cannot be nonradiative.

Superconvergence recovery of cubic edge elements for Maxwell’s equations

In this article, a new recovery method is designed and analyzed for curl conforming elements on cuboid mesh. The proposed recovery method fully exploits the potential of symmetry to obtain the superconvergence of recovered edge finite element solution in discrete ℓ2 norm. The idea is to identify a symmetry sub-domain such that a polynomial of the same degree is recovered by local L2 projection, based on which the recovered values follow. Combined with extrapolation and the least square fitting for the linear edge elements and quadratic edge elements, respectively, we obtain global superconvergence for recovered quantities in L2 norm. Numerical experiments are provided to validate our theoretical findings.