In this paper, a topology optimization algorithm for the mechanical-electrical coupling problem of periodic composite materials is studied. Firstly, the homogenization problem of the mechanical-electrical coupling topology optimization problem of periodic composite materials is established by the multi-scale asymptotic expansion method. Secondly, the topology optimization algorithm for the mechanical-electrical coupling problem of periodic composite materials is constructed by finite element method, solid isotropic material with penalisation method and homogenization method. Finally, numerical results show that the proposed algorithm is effective to calculate the optimal structure of the periodic composite cantilever beam under the influence of the mechanical-electrical coupling.

In this paper, we mainly study the high-order numerical scheme of right Caputo time fractional differential equations with uniform accuracy. Firstly, we construct the high-order finite difference method for the right Caputo fractional ordinary differential equations (FODEs) based on piecewise quadratic interpolation. The local truncation error of right Caputo FODEs is given, and the stability analysis of the right Caputo FODEs is proved in detail. Secondly, the time fractional partial differential equations (FPDEs) with right Caputo fractional derivative is studied by coupling the time-dependent high-order finite difference method and the spatial central second-order difference scheme. Finally, three numerical examples are used to verify that the convergence order of high-order numerical scheme is $3-\lambda$ in time with uniform accuracy.

We give a modified block-by-block method for the nonlinear fractional order Volterra integral equation system by using quadratic Lagrangian interpolation based on the classical block-by-block method. The core of the method is that we divide its domain into a series of subdomains, that is, block it, and use piecewise quadratic Lagrangian interpolation on each subdomain to approximate $\mathit{\kappa }(x,y,s,r,u(s,r))$. Our proposed method has uniform accuracy and its convergence order is $O(h_x^{4-\alpha }+h_y^{4-\beta })$. We give a strict proof for the error analysis of the method, and give several numerical examples to verify the correctness of the theoretical analysis.

In this paper, we proposed a higher-order uniform accuracy scheme for nonlinear $\psi$-Volterra integral equations in two dimension with weakly singular kernel by using the modified block-by-block method. First, we constructed a high order uniform accuracy scheme method in this paper by dividing the entire domain into some small sub-domains and approximating the integration function with biquadratic interpolation in each sub-domain. Second, we rigorously proved that the convergence order of the higher order uniform accuracy scheme was $O(h_s^{3+{\sigma }_1}+h_t^{3+{\sigma }_2})$ with $0<{\sigma }_1,{\sigma }_2<1$ by using the discrete Gronwall inequality. Finally, two numerical examples were used to illustrate experimental results with different values of $\psi$ to support the theoretical results.