Time-varying features are generally considered to be detrimental to the analysis and design of control systems. This paper establishes methods to design bounded linear time-varying (LTV) controllers ...such that the control performance of a linear time-invariant (LTI) system can be improved, that is, the finite-time stability of the closed-loop system can be obtained. Specifically, for an LTI control system, by using the solution to a parametric Lyapunov equation (PLE), a bounded LTV controller containing a suitable time-varying parameter is designed. By fully exploiting properties of the solution to the PLE, it is shown that the closed-loop system is finite-time stable. Both state feedback and observer based output feedback, in which both the observer gain and the state feedback gain are time-varying, are considered. As a consequence, the finite-time semi-global stabilization and the fixed-time (prescribed finite-time) stabilization problems for linear systems by bounded controls are solved. The established method is utilized to the design of the spacecraft rendezvous control system and its effectiveness is verified by simulations.
This paper enfolds a medius analysis for the Stokes equations and compares different finite element methods (FEMs). A first result is a best approximation result for a P1 non-conforming FEM. The main ...comparison result is that the error of the P2P0-FEM is a lower bound to the error of the Bernardi–Raugel (or reduced P2P0) FEM, which is a lower bound to the error of the P1 non-conforming FEM, and this is a lower bound to the error of the MINI-FEM. The paper discusses the converse direction, as well as other methods such as the discontinuous Galerkin and pseudostress FEMs.
Furthermore this paper provides counterexamples for equivalent convergence when different pressure approximations are considered. The mathematical arguments are various conforming companions as well as the discrete inf-sup condition.
This paper presents advanced computer simulation of rock cutting process typical for excavation works in civil engineering. Theoretical formulation of the hybrid discrete/finite element model has ...been presented. The discrete and finite element methods have been used in different subdomains of a rock sample according to expected material behaviour, the part which is fractured and damaged during cutting is discretized with the discrete elements while the other part is treated as a continuous body and it is modelled using the finite element method. In this way, an optimum model is created, enabling a proper representation of the physical phenomena during cutting and efficient numerical computation. The model has been applied to simulation of the laboratory test of rock cutting with a single TBM (tunnel boring machine) disc cutter. The micromechanical parameters have been determined using the dimensionless relationships between micro- and macroscopic parameters. A number of numerical simulations of the LCM test in the unrelieved and relieved cutting modes have been performed. Numerical results have been compared with available data from in-situ measurements in a real TBM as well as with the theoretical predictions showing quite a good agreement. The numerical model has provided a new insight into the cutting mechanism enabling us to investigate the stress and pressure distribution at the tool–rock interaction. Sensitivity analysis of rock cutting performed for different parameters including disc geometry, cutting velocity, disc penetration and spacing has shown that the presented numerical model is a suitable tool for the design and optimization of rock cutting process.
A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed ...by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons/polyhedra. The WG-MFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. Allowing the use of discontinuous approximating functions on arbitrary shape of polygons/polyhedra makes the method highly flexible in practical computation. Optimal order error estimates in both discrete H^1 and L^2 norms are established for the corresponding weak Galerkin mixed finite element solutions.
Structural finite-element analysis (FEA) has been widely used to study the biomechanics of human tissues and organs, as well as tissue–medical device interactions, and treatment strategies. However, ...patient-specific FEA models usually require complex procedures to set up and long computing times to obtain final simulation results, preventing prompt feedback to clinicians in time-sensitive clinical applications. In this study, by using machine learning techniques, we developed a deep learning (DL) model to directly estimate the stress distributions of the aorta. The DL model was designed and trained to take the input of FEA and directly output the aortic wall stress distributions, bypassing the FEA calculation process. The trained DL model is capable of predicting the stress distributions with average errors of 0.492% and 0.891% in the Von Mises stress distribution and peak Von Mises stress, respectively. This study marks, to our knowledge, the first study that demonstrates the feasibility and great potential of using the DL technique as a fast and accurate surrogate of FEA for stress analysis.
In this review paper, we are mainly concerned with the finite difference methods, the Galerkin finite element methods, and the spectral methods for fractional partial differential equations (FPDEs), ...which are divided into the time-fractional, space-fractional, and space-time-fractional partial differential equations (PDEs). Besides, fast algorithms for the FPDEs are included in order to stimulate more efficient algorithms for high-dimensional FPDEs.
We present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional ...Poisson problem. Relying on the equilibration by the mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree. Maximal local overestimation is guaranteed as well. Numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous Galerkin scheme.
In this article, we propose an unconventional method, i.e., generalized finite difference method (GFDM), to analyze the electromagnetic performance of surface-mounted permanent magnet (SPM) machine ...with eccentric rotor shape. The remarkable characteristic of this method is that it is meshless. In this method, the solution region is discretized in the form of nodes, and the nodes can move with the moving structure. This is very helpful in modeling the SPM machine because there is no remeshing process during the rotation. However, mesh generation is required in the traditional finite-element analysis (FEA). First, the basic principle of the proposed GFDM is introduced. Second, the whole field domain is divided into five types of regions, viz., PMs, air gap, slot openings, slots, and stator core. Third, the differential equations satisfied by each region are discretized into algebraic equations by the GFDM. The interfaces need to be dealt with separately because of the discontinuity of the derivative of the vector potential. Then, the systems of equations are solved to analyze the electromagnetic performance of an SPM machine. Finally, this method is implemented on a 12-slot, 10-pole SPM machine. Meanwhile, the effectiveness of this method is verified by an FEA and experiment.
We consider a system of fully nonlinear partial differential equations that corresponds to the Hamilton–Jacobi–Bellman equations for the value functions of an optimal innovation investment problem of ...a monopoly firm facing bankruptcy risk. We compare several algorithms for the numerical solution of the considered problem: the collocation method, the finite difference method, WENO method and the adaptive finite element method. We discuss implementation issues for the considered schemes and perform numerical studies for different model parameters to assess their performance.
PDF
