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Programming the shape of soft smart materials is a challenging task due to the enormous design space involved. In this study, we propose a novel approach to determine applied stimuli that enable the desired actuated shapes of soft smart materials. Our approach combines finite element methods (FEM), deep neural networks (DNN), and particle swarm optimization (PSO). By employing beams made of dielectric elastomer (DE) as models, we partition a DE beam into multiple actuating units, allowing independent actuation through the application of paired electrical stimuli. FEM is subsequently employed to compute the actuated shapes in response to various applied stimuli. The selection of these stimuli is accomplished through the utilization of Latin hypercube sampling. By utilizing the FEM-derived data, we have developed a machine learning (ML) surrogate model that integrates a long short-term memory (LSTM) network with a fully connected neural network (FCNN). Finally, PSO is employed to determine the optimal applied stimulus that yields the desired actuated shape. The LSTM-FCNN surrogate model is utilized to evaluate the fitness of PSO. The ML-PSO framework exhibits remarkable performance and efficiency in the context of the inverse design of soft smart beams. [Display omitted] • Nonlinear FEM is employed to accurately capture the relationship between the actuated shape and external stimuli of dielectric elastomers. • Machine learning surrogate models enable fast and highly accurate prediction of shape morphing. • The combination of particle swarm optimization and machine learning proves to be efficient in designing the shape of soft smart beams. • This approach can be applied to control soft beam robots under various types of stimuli. [ABSTRACT FROM AUTHOR]

Live working in distribution work is an important way to ensure normal power supply of power equipment. Live installation tool for robot operation was proposed in this paper in order to reduce the safety accidents occurrence. Through different structures and sensor combinations, it can complete the autonomous operation by cooperating with robot, including lead wire monitoring, line detection and tightening the puncture clamp. Finally, the finite element analysis of gear transmission was carried out, and the results showed that the design of the tool can mees the strength requirements. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the parallel two-grid finite element method for the transient natural convection problem with non-smooth initial data. Our numerical scheme involves solving a nonlinear natural convection problem on the coarse grid and solving a linear natural convection problem on the fine grid. The linear natural convection problem can be split into two subproblems which can be solved in parallel: a linearized Navier–Stokes problem and a linear parabolic problem. We firstly provide the stability and convergence of standard Galerkin finite element method with non-smooth initial data. Secondly, we develop optimal error estimates of two-grid finite element method for velocity and temperature in H 1 -norm and for pressure in L 2 -norm. In order to overcome the difficulty posed by the loss of regularity, some suitable weight functions are introduced in our stability and convergence analysis for the natural convection equations. Finally, some numerical results are presented to verify the established theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations is considered with nonsmooth initial data. Our numerical scheme is based on the implicit Crank–Nicolson scheme for the linear terms and the explicit Adams–Bashforth scheme for the nonlinear terms for the temporal discretization, standard Galerkin finite element method is used to the spatial discretization. In order to improve the computational efficiency, the decoupled method is introduced, as a consequence the original problem is split into two linear subproblems, and these subproblems can be solved in parallel. We verify that our numerical scheme is almost unconditionally stable for the nonsmooth initial data ( u 0 , θ 0 ) with the divergence-free condition. Furthermore, under some stability conditions, we show that the error estimates for velocity and temperature in L 2 norm is of the order O ( h 2 + Δ t 3 2 ) , in H 1 norm is of the order O ( h 2 + Δ t ) , and the error estimate for pressure in a certain norm is of the order O ( h 2 + Δ t ) . Finally, some numerical examples are provided to verify the established theoretical findings and test the performances of the developed numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stability and convergence results of numerical solutions in fully discrete fractional-step formulation for the Oldroyd model. The proposed numerical method is constructed by the decompositions of the viscosity in time part and the finite element method in space part. With some mild regularity assumptions on the exact solution, the unconditional stability results of the approximate solutions are established. Then, the first order spatial convergence for the “end-of-step” velocity is shown. Based on the above results, the optimal order approximations for the velocity and pressure in L 2 -norm are obtained for the mesh size. Finally, two numerical examples are given to illustrate the established theoretical analysis and test the performances of the developed numerical method and show the influences of the memory term for the considered problem. [ABSTRACT FROM AUTHOR]

In this paper, the H 2 -stability of the first order fully discrete Galerkin finite element methods for the Boussinesq equations with smooth and non-smooth initial data is presented. The finite element spatial discretization for the Boussinesq equations is based on the mixed finite element method, and the temporal treatments of the spatial discrete Boussinesq equations include the implicit scheme, the semi-implicit scheme, the implicit/explicit scheme and the explicit scheme. The H 2 -stability results of the above numerical schemes are established. Firstly, we prove that the implicit and semi-implicit schemes are the H 2 -unconditional stable. Then we show that the implicit/explicit scheme is H 2 -almost unconditional stable with the initial data that belong to H 1 and H 2 , and the similar results are obtained for the semi-implicit/explicit scheme in the case of the initial data that belong to L 2 . Furthermore, we show that the explicit scheme is the H 2 -conditional stable. Finally, some numerical examples are provided to verify the established theoretical findings and confirm the corresponding H 2 stability analysis of the different numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the posteriori error estimates of Galerkin finite element method for the unsteady Navier–Stokes equations. By constructing the approximate Navier–Stokes reconstructions, the errors of velocity and pressure are split into two parts. For the estimates of time part, the energy method and other skills are used, for the estimates of spatial part, the well-developed theoretical analysis of posteriori error estimates for the elliptic problem can be adopted. More important, the error estimates of time part can be controlled by the estimates of spatial part. As a consequence, the posteriori error estimates in L ∞ (0, T ; L 2 (Ω)),  L ∞ (0, T ; H 1 (Ω)) and L 2 (0, T ; L 2 (Ω)) norms for velocity and pressure are derived in both spatial discrete and time-space fully discrete schemes. [ABSTRACT FROM AUTHOR]

In this paper, the decoupled stabilized finite element methods are developed for the Boussinesq equations with temperature dependent coefficients. In our numerical schemes, the original problem is decoupled into the linearized Navier-Stokes equations and a parabolic problem, the low order mixed finite element pairs are used to approximate the spatial spaces and the backward Euler scheme is adopted to treat the time terms. Furthermore, the linear terms are dealt with the implicit scheme while the nonlinear terms are treated by the semi-implicit scheme, then, a lot of storage and computational cost are saved. The advantages for our numerical schemes are parameter-free, unconditionally stable, and constant matrix at each time level. Finally, some numerical tests are presented to verify the theoretical results of the developed numerical methods, and show that our schemes not only keep good accuracy but also save a lot of computational times. [ABSTRACT FROM AUTHOR]

• This paper considers the first order semi-implicit energy stable finite element method for the two phase magnetohydrodynamic flows (the Cahn–Hilliard equation coupling the incompressible MHD system), some numerical results are provided to show the performances of the considered numerical scheme. The main contribution of this work is to present the rigorous unconditional stability of numerical solutions in both time semi-discrete and fully discrete cases with two suitable stabilized terms. Optimal error estimates of numerical solutions are also presented by using the energy method and the Gronwall lemma. Our findings enrich and develop the theoretical analysis results of mixed finite element for the two-phase incompressible flows. This paper is devoted to present the rigorous unconditional stability and optimal error estimates of first order semi-implicit energy stable finite element method developed by Yang et al. (Comput. Methods Appl. Mech. Engrg. 356 (2019) 435–464) for the two phase magnetohydrodynamic(MHD) flows, some numerical results are also provided to show the performances of the considered numerical scheme. [ABSTRACT FROM AUTHOR]

For the time-dependent thermal convection problems with temperature-dependent coefficients, the implicit–explicit scheme is presented, in which mixed finite element method is applied for the spatial approximation of the velocity, pressure and temperature while the time discretization is based on the high-order backward difference scheme. Linear terms are dealt with the implicit scheme while the nonlinear terms are treated by the semi-implicit scheme. The advantages for this scheme are unconditionally stable, decoupled computational and second order accuracy. Finally, numerical tests illustrate the theoretical results of the presented schemes, and display that the highly efficient method conserves the property of divergence free of the original problems. [ABSTRACT FROM AUTHOR]

In the paper we consider the penalty finite element method for the stationary incompressible magnetohydrodynamics (MHD) problem with a factor of penalty parameter. Stability and convergence of numerical solutions are established. Furthermore, two level penalty methods are also developed for the MHD problem. Our methods consist of solving a nonlinear MHD problem by the usual penalty method on a coarse mesh with mesh size H , and then a linearized MHD problem based on the Stokes, Newton and Oseen iterations on a fine mesh is solved by the penalty method with mesh size h ( h ≪ H ) , respectively. Stability and error estimates of numerical solutions in two level penalty methods are presented. Finally, some numerical tests are provided to demonstrate the effectiveness of the developed algorithms. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we develop a posteriori error estimates for the steady Navier–Stokes equations based on the lowest equal-order mixed finite element pair. Residual type a posteriori error estimates are derived by means of general framework established by Verfürth for the nonlinear equations. Furthermore, a simple error estimator in norm is also presented by using the duality argument. Numerical experiments using adaptive computations are presented to demonstrate the effectiveness of these error estimates for three examples. The first example is a singular problem with known solution, the second example is a physical model of lid driven cavity and the last one is a backward facing step problem. [Copyright &y& Elsevier]

In this paper, the stability and convergence results of finite volume method for the time-dependent incompressible magnetohydrodynamics equations are developed based on triangulation. The linear polynomials are used to approximate the velocity, magnetic fields and pressure. In order to overcome the restriction of the discrete inf-sup condition, the local pressure projection method is employed. Firstly, two self-adjoint linear mappings are introduced to establish the equivalences of temporal terms and bilinear terms in finite element method and finite volume method. Secondly, the existence and uniqueness of numerical solutions are presented by using the Oseen iteration and mathematical induction method. Thirdly, H 2 -stability and optimal error estimates are provided by introducing the Galerkin projection of generalized bilinear terms in finite volume method, using the energy method and constructing the corresponding dual problem. The main theoretical findings of this paper are as follows: ‖ (u h (t) , H h (t)) ‖ 0 2 + ∫ 0 t (min ⁡ { ν , γ σ − 1 } ‖ (u h , H h) ‖ 1 2 + G (p h , p h)) d s ≤ C , min ⁡ { ν , γ σ − 1 } ‖ (u h (t) , H h (t)) ‖ 1 2 + G (p h (t) , p h (t)) + ∫ 0 t ‖ (u h t , H h t) ‖ 0 2 d s ≤ C , ‖ ∇ (u h , H h) ‖ 0 2 + ∫ 0 t min ⁡ { ν , γ σ − 1 } ‖ (A 1 h u h , A 2 h H h) ‖ 0 2 d s ≤ C , τ 1 2 (t) ‖ (u (t) − u h (t) , H (t) − H h (t)) ‖ 0 + h (‖ (u (t) − u h (t) , H (t) − H h (t)) ‖ 1 + τ 1 2 (t) ‖ p (t) − p h (t) ‖ 0) ≤ C h 2 , where τ (t) = min ⁡ { 1 , t } , the letters ν , σ are the physical parameters, C and γ are two positive constants independent of the mesh size h , (u , H , p) and (u h , H h , p h) are the solutions of problems (2.1) and (4.1). Finally, some numerical results are presented to verify the established theoretical findings and show the performances of the considered numerical method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a fully discrete defect-correction mixed finite element method (MFEM) for solving the non-stationary conduction–convection problems in two dimension, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. In this method, we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct these solutions on the same grid using a linearized defect-correction technique. The stability and the error analysis are derived. The theory analysis shows that our method is stable and has a good convergence property. Some numerical results are also given, which show that this method is highly efficient for the unsteady conduction–convection problems. [Copyright &y& Elsevier]

Abstract: In this article, new one- and two-level Newton iterative mixed finite element methods (MFEM) for two-dimensional stationary conduction–convection problems are presented. In both of the methods, we decouple the conduction–convection problem into two systems: one is for the temperature and the other is for the velocity and pressure. Thus much computational time can be saved. In the new Newton MFEM, we first solve the equation for the temperature, then solve a Navier–Stokes problem by Newton iterative method. While in the two-level method, we solve the decoupled conduction–convection problem as same as in the one-level method using a coarse grid firstly, then seek a fine grid solution by solving a linearized problem on a fine grid. Stability analysis is performed and error estimates of the methods are derived, which show that our methods are stable and have a good precision. Numerical experiments are also given, which confirm the theoretical analysis and demonstrate the efficiency of the new methods. [ABSTRACT FROM AUTHOR]

In this paper, we develop an online generalized multiscale approximation of the multipoint flux mixed finite element method (MFMFE) for Darcy flow in highly heterogeneous porous media, which can handle full tensor permeability fields and unstructured grids with local velocity elimination. Here, for illustration of the proposed method, we will mainly consider the situation that the porous medium is filled with fractures, which usually results in unstructured grids, and the extension to the situation that the permeability is represented as a full tensor is straightforward. From the fine-grid discretization by the MFMFE method, we derive the symmetric and positive definite discrete bilinear form for both the matrix and fracture pressure, and thereby obtain the corresponding discrete weak formulation that only related to pressure. In the offline stage, we compute offline basis functions which contain the important local multiscale information of each coarse-grid block to form the initial multiscale space. In the online stage, we construct and add online basis functions to enrich the multiscale space and consequently improve the accuracy of the multiscale solution. Each offline basis function and each online basis function contains multiscale information both for matrix and fracture. We give the theoretical analysis for the convergence of the online enrichment, which shows that more sufficient initial basis functions lead to faster convergence rates. A series of numerical examples that either the permeability is a full tensor or the porous medium is filled with fractures are presented to show the performance of the multiscale method. • An online generalized multiscale approximation of the MFMFE method is proposed. • The proposed method can handle full tensor permeabilities and unstructured grids. • The proposed method can simulate Darcy flow in fractured porous medium. • The discrete bilinear form for pressure is symmetric and positive definite. [ABSTRACT FROM AUTHOR]

Various disturbances do exist in the image taking process of scanning transmission electron microscopes (STEM), which seriously reduces the resolution and accuracy of STEM images. In this paper, a deep understanding of the scanning distortion influence on the real and reciprocal spaces of STEM images is achieved via theoretical modeling and simulation. A scanning distortion correction algorithm is further developed based on two images scanned along perpendicular directions, which is able to effectively correct scanning distortion induced deviations and significantly increase the signal to noise ratio of STEM images. [ABSTRACT FROM AUTHOR]

Abstract: Indentation stresses in single- and multi-layer delaminated thin films made of elastic–perfectly plastic materials in microwedge indentation delamination tests are analyzed via finite element calculations with different wedge angles and other geometrical and mechanical parameters. Based on the formula for a single-layer thin film under indentation loading [Zhao et al. J Mater Res 2009;24:1943] and by introducing the equivalent material parameters, we developed simple analytical formulae for the loading indentation stress (as well as the energy release rate) in each layer of the multi-layer thin film, in terms of the residual stress, elastic modulus, Poisson’s ratio, yielding strength, and thickness of each layer. Our analytical solution is validated by the finite element calculations and should be useful to thin-film delamination tests. [Copyright &y& Elsevier]