This paper illustrates the efficacy of using accelerated gradient descent schemes for minimizing a uniaxially constrained Landau-de Gennes model for nematic liquid crystals. Three (alternating direction) minimization schemes are applied to a structure preserving finite element discretization of the uniaxial model: a standard gradient descent method, the "heavy-ball" method, and Nesterov's method. The performance of the schemes is measured in terms of the number of iterations required to obtain the equilibrium state, as well as the total computational time (wall time).

The numerical experiments clearly show that the accelerated gradient descent schemes reduce the number of iterations and computational time significantly, despite the hard uniaxial constraint that is not "smooth'' when defects are present. Moreover, our results show that accelerated schemes are not hindered when combined with an alternating direction minimization algorithm and are easy to implement.

In this paper, spectral approximations for distributed optimal control problems governed by the Stokes equation are considered. And the constraint set on velocity is stated with $L^2$-norm. Optimality conditions of the continuous and discretized systems are deduced with the Karush-Kuhn-Tucker conditions and a Lagrange multiplier depending on the constraint. To solve the equivalent systems with high accuracy, Galerkin spectral approximations are employed to discretize the constrained optimal control systems. Meanwhile, we adopt a parameter $\lambda$ in the pressure approximation space, which also guarantees the inf-sup condition, and study a priori error estimates for the velocity and pressure. Specially, an efficient algorithm based on the Uzawa algorithm is proposed and its convergence results are investigated with rigorous analyses. Numerical experiments are performed to confirm the theoretical results.

In this paper, we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations (RSFDEs) with time delay, which constitute an important class of differential equations of practical significance. We develop a novel implicit alternating direction method that can effectively and efficiently tackle the RSFDEs in both two and three dimensions. The numerical method is proved to be uniquely solvable, stable and convergent with second order accuracy in both space and time. Numerical results are presented to verify the accuracy and efficiency of the proposed numerical scheme.

Deformable microcapsules are widely used in industries and also serve as a mechanical model of living biological cells. In this study, we develop a novel method, by integrating a deep convolutional neural network (DCNN) with high-fidelity mechanistic capsule modelling, to identify the membrane constitutive law and estimate associated parameters of a microcapsule from its steady deformed profile in a capillary tube. Compared with conventional inverse methods, the present approach is more accurate and can increase the prediction throughput rate by a few orders of magnitude. It can process capsules with large deformation in inertial flows. Furthermore, the method can predict the capsule membrane shear elasticity, area dilatation modulus and initial inflation from a single steady capsule profile. We explore the mechanism that the DCNN makes decisions by considering its feature maps, and discuss their potential implication on the development of inverse methods. The present method provides a promising tool which may enable high-throughput mechanical characterisation of microcapsules and biological cells in microfluidic flows.

We develop a new conservative Allen-Cahn phase-field model for diblock copolymers in this paper by using the Allen-Cahn type gradient flow approach for the classical Ohta-Kawaski free energy. The change in volume fraction of two composing monomers is eliminated by using a nonlocal Lagrange multiplier. Based on the recently developed stabilized Scalar Auxiliary Variable method, we have further developed an effective numerical scheme to solve the model. The scheme is highly efficient and only two linear and decoupled equations are needed to solve at  every time step. We then prove that the numerical  method is unconditionally energy stable, the stability and accuracy of the new scheme are  demonstrated by numerous numerical examples conducted. By qualitatively comparing the equilibrium solution obtained by the new model and the classic Cahn-Hilliard model, we illustrate the effectiveness of the new model.

Calculating interacting stress intensity factors (SIFs) of multiple elliptical-holes and cracks is very important for safety assessment, stop-hole optimization design and resource exploitation production in underground rock engineering, e.g., buried tunnels, deep mining, geothermal and shale oil/gas exploitation by hydraulic fracturing technology, where both geo-stresses and surface stresses are applied on buried tunnels, horizontal wells and natural cracks. However, current literatures are focused mainly on study of interacting SIFs of multiple elliptical-holes (or circular-holes) and cracks only under far-field stresses without consideration of arbitrary surface stresses. Recently, our group has proposed a new integral method to calculate interacting SIFs of multiple circular-holes and cracks subjected to far-filed and surface stresses. This new method will be developed to study the problem of multiple elliptical-hole and cracks subjected to both far-field and surface stresses. In this study, based on Cauchy integral theorem, the exact fundamental stress solutions of single elliptical-hole under arbitrarily concentrated surface normal and shear forces are derived to establish new integral equation formulations for calculating interacting SIFs of multiple elliptical-holes and cracks under both far-field and arbitrary surface stresses. The new method is proved to be valid by comparing our results of interacting SIFs with those obtained by Green's function method, displacement discontinuity method, singular integral equation method, pseudo-dislocations method and finite element method. Computational examples of one elliptical-hole and one crack in an infinite elastic body are given to analyze influence of loads and geometries on interacting SIFs. Research results show that when $\sigma_{x x}^{\infty} \geq \sigma_{y y}^{\infty}$, there appears a neutral crack orientation angle $\beta_0$ (without elliptical-hole's effect). Increasing $\sigma_{x x}^{\infty} / \sigma_{y y}^{\infty}$  and $b/a$ (close to circular-hole) usually decreases $\beta_0$ of $K_I$ and benefits to the layout of stop-holes. The surface compressive stresses applied onto elliptical-hole $(n)$ and crack $(p)$ have significant influence on interacting SIFs but almost no on $\beta_0$. Increasing $n$ and $p$ usually results in increase of interacting SIFs and facilitates crack propagation and fracture networks. The elliptical-hole orientation angle $(\alpha)$ and holed-cracked distance $(t)$ have great influence on the interacting SIFs while have little effect on $\beta_0$. The present method is not only simple (without any singular parts), high-accurate (due to exact fundamental stress solutions) and wider applicable (under far-field stresses and arbitrarily distributed surface stress) than the common methods, but also has the potential for the anisotropic problem involving multiple holes and cracks.

Ultrasound is a low-cost, non-invasive and real-time imaging modality that has proved popular for many medical applications. Unfortunately, the acquired ultrasound images are often corrupted by speckle noise from scatterers smaller than ultrasound beam wavelength. The signal-dependent speckle noise makes visual observation difficult. In this paper, we propose a patch-based low-rank approach for reducing the speckle noise in ultrasound images. After constructing the patch group of the ultrasound images by the block-matching scheme, we establish a variational model using the weighted nuclear norm as a regularizer for the patch group. The alternating direction method of multipliers (ADMM) is applied for solving the established nonconvex model. We return all the approximate patches to their original locations and get the final restored ultrasound images. Experimental results are given to demonstrate that the proposed method outperforms some existing state-of-the-art methods in terms of visual quality and quantitative measures.

A stabilizer-free weak Galerkin finite element method is proposed for the Stokes equations in this paper. Here we omit the stabilizer term in the new method by increasing the degree of polynomial approximating spaces for the weak gradient operators. The new algorithm is simple in formulation and the computational complexity is also reduced. The corresponding approximating spaces consist of piecewise polynomials of degree $k\geq1$ for the velocity and $k-1$ for the pressure, respectively. Optimal order error estimates have been derived for the velocity in both $H^1$ and $L^2$ norms and for the pressure in $L^2$ norm. Numerical examples are presented to illustrate the accuracy and convergency of the method.

This paper mainly considers the optimal convergence analysis of the $\theta$--Maruyama method for stochastic Volterra integro-differential equations (SVIDEs) driven by Riemann--Liouville fractional Brownian motion under the global Lipschitz and linear growth conditions. Firstly, based on the contraction mapping principle, we prove the well-posedness of the analytical solutions of the SVIDEs. Secondly, we show that the $\theta$--Maruyama method for the SVIDEs can achieve strong first-order convergence. In particular, when the $\theta$--Maruyama method degenerates to the explicit Euler--Maruyama method, our result improves the conclusion that the convergence rate is $H+\frac{1}{2},$ $ H\in(0,\frac{1}{2})$ by Yang et al., J. Comput. Appl. Math., 383 (2021), 113156. Finally, the numerical experiment verifies our theoretical results.

In this article, an analytical method is proposed to analyze of the linear buckling behavior of the FG porous truncated conical shells subjected to a uniform axial compressive load and resting on the Pasternak elastic foundation. The material properties including Young's modulus, shear modulus and density are assumed to vary in the thickness direction. Three types of FG porous distributions including symmetric porosity distribution, non-symmetric porosity and uniform porosity distribution are considered. The governing equations of the FG porous truncated conical shells are obtained by using the first-order shear deformation theory (FSDT). With the help of the Galerkin method, the expressions for critical buckling loads are obtained in closed forms. The reliability of the obtained results is verified by comparing the present solutions with the published solutions. Finally, the numerical results show the effects of shell characteristics, porosity distribution, porosity coefficient, and elastic foundation on the critical buckling load.

In this paper, the transient Navier-Stokes equations with damping are considered. Firstly, the semi-discrete scheme is discussed and optimal error estimates are derived. Secondly, a linearized backward Euler scheme is proposed. By the error split technique, the Stokes operator and the $H^{-1}$-norm estimate, unconditional optimal error estimates for the velocity in the norms ${L^\infty}(L^2)$ and ${L^\infty}(H^1)$, and the pressure in the norm ${L^\infty}(L^2)$ are deduced. Finally, two numerical examples are provided to confirm the theoretical analysis.

In this work, we have developed a fifth-order alternative mapped weighted essentially nonoscillatory (AWENO-M) finite volume scheme using non-linear weights of mapped WENO reconstruction scheme of Henrick et al. (J. Comput. Phys., 207 (2005), pp. 542--567) for solving hyperbolic conservation laws. The reconstruction of numerical flux is done using primitive variables instead of conservative variables. The present scheme results in less spurious oscillations near discontinuities and shows higher-order accuracy at critical points compared to the alternative WENO scheme (AWENO) based on traditional non-linear weights of Jiang and Shu (J. Comput. Phys., 228 (1996), pp. 202--228). The third-order Runge-Kutta method has been used for solution advancement in time. The Harten-Lax-van Leer-Contact (HLLC) shock-capturing method is used to provide necessary upwinding into the solution. The performance of the present scheme is evaluated in terms of accuracy, computational cost, and resolution of discontinuities by using various one and two-dimensional test cases.