Component design (“digitalization”), performance optimization (“optimization”), and process simulation (“simulation”) are three critical modules in the 3D printing process. “Digitalization” refers to the transformation of design drawings or pre-processed physical objects into editable digital components through means such as images, videos, and scanning processes. “Optimization” involves the application of constraints from physical fields such as mechanics and thermodynamics to enhance the performance of digital components. “Simulation” entails the digital simulation and twin modeling of physicochemical changes during the manufacturing process, based on the optimized components, to emulate real-world physical conditions. This research aims to introduce integrated modeling and algorithmic studies in design, optimization, and simulation within the phase field framework. In the “digitalization” module, we will present three-dimensional reconstruction models, repair models, and lightweight support structure design models that correspond to common data types in computer-aided design. In the “optimization” module, we will introduce a series of multi-scale, multi-physical field, and multi-material coupled topology optimization problems and their corresponding solutions. In the “simulation” module, we will discuss macroscopic (phase transition) to microscopic (grain boundary) scale coupling theories and the physical field couplings such as laser-thermal-flow-solid, in addition to simulating methods for processes like Fused Deposition Modeling (FDM) and Selective Laser Melting (SLM), integrating 3D printing parameters and techniques. The integrated research approach to design, optimization, and simulation based on the phase field model framework aims to shorten the product development cycle, enhance design efficiency, and provide a theoretical basis and algorithmic support for quality traceability and root cause analysis of 3D printed components.

Unconventional reservoirs, such as shale/tight oil and gas reservoirs, are typically multi-scale spaces with both nano- and centimetre-scale matrix porosity, micrometre-to-centimetre-scale natural fracture systems and metre-to-hundred-metre-scale man-made fractures resulting from large-scale fracturing. In this paper, we present a mathematical model of the coupled fluid flow over matrix, micro-fractures, macroscopic fractures, and large-scale fracture cavities in fractured porous media.

In 2016, Karney proposed an exact sampling algorithm for the standard normal distribution. In this article, we present an exact sampling algorithm for the normal distribution of standard variance $\sqrt{1/(2\ln2)}$ and mean $0$, which can be called the binary Gaussian distribution, as its relative probability density function is given by $2^{-x^2}$ for $x\in\mathbb{R}$. Our proposed algorithm requires no floating-point arithmetic in practice, and can be regarded as the promotion of Karney's exact sampling technique. We give an estimate of the expected number of uniform deviates over the range $(0,1)$ used by this algorithm until outputting a sample value. Numerical experiments also demonstrate the effectiveness of the sampling algorithm. For any rational number $c$ greater than $1$ but less than Euler's number $e$, the idea of sampling exactly the binary Gaussian is generalized to a class of normal distributions of standard variance $\sqrt{1/(2\ln{c})}$ and mean $0$, called “$c$-ary Gaussian distributions”, and a similar complexity analysis is presented.

The physical quantities in the electronic structure calculation of periodic systems involve the band integral over the Brillouin zone. For metals, the integrand is discontinuous when the Fermi surface crosses through the energy band, which makes it difficult to improve the calculation accuracy of general numerical integration methods. Based on smearing methods, we propose two numerical formats with higher order precision about the broadening parameters by first and second extrapolation respectively. The numerical method based on this kind of extrapolation scheme gives more precise integral approximation over the Brillouin zone with the premise of the same discrete $\mathbf{k}$-point in Brillouin region, and significantly improves the computational efficiency. The efficiency of the extrapolation methods is verified by numerical tests on two typical systems.

In this paper, we develop some numerical schemes to solve fractional nonlinear Schröd\-inger equation, which preserve one or more analytical properties of the fractional system. First, we apply BDF scheme, Crank-Nicolson scheme and relaxation scheme to discrete time derivative, and analytic conservation and dispersion error of the discrete schemes. Second, we use central difference scheme and compact difference scheme to discrete space fractional derivative of the fractional nonlinear Schrödinger equation with periodic boundary condition. We find that central difference scheme and compact difference scheme preserve mass and energy conservation laws very well for periodic boundary condition. Finally, the numerical experiments of some fractional nonlinear Schrödinger equations are given to verify the correctness of theoretical results.

This paper adopts the Newmark integration method based on the large-scale tearing finite element method to perform high-precision large-scale parallel solving of structural dynamic calculations. A multi-level load balancing strategy combining static and dynamic methods is designed for heterogeneous platforms. For inter-node computing, subdomain boundaries are partitioned based on the characteristics of the tearing finite element method, and a domain boundary balanced graph bipartition algorithm is used to balance the computation load of each subdomain. For intra-node computing, dynamic optimization of computation load is performed based on the performance differences of computing units on heterogeneous platforms. To improve the utilization rate of heterogeneous computing platforms, multi-stream parallel optimization is carried out for the core computing module's batch matrix-vector multiplication. The optimization in this paper has been integrated into the high-performance numerical simulation software for structural mechanics, HARSA-feti. The simulation performance is demonstrated using the flow-induced vibration simulation of a real reactor fuel component as an example. The results show that the simulation performance has increased by more than 71.3\%, and the high-precision simulation of a billion-grid-scale full-core fuel rod component has been achieved for the first time. Compared with 1, 000 GPUs, the strong and weak scalable parallel efficiency of 16, 000 GPUs reached 74.1\% and 81.1\%, respectively.

Three explicit Runge-Kutta schemes are constructed from the viewpoint of Newton-Cotes integration, including a 3-stage $2^{nd}$ order Runge-Kutta scheme with parameters and two 4-stage $3^{rd}$ order Runge-Kutta schemes with parameters. The commonly used RK3 and RK4 can be obtained by taking special parameters from our schemes. The accuracy and the stability of these schemes are presented. Numerical examples are given to verify the stability, effectiveness and high accuracy of the constructed schemes. Results show that our explicit Runge-Kutta schemes have better stability than that of the classic explicit Runge-Kutta schemes.

High-dimensional data set arises in many fields, which means the feature size is greater than or far greater than the sample size. To deal with these, there have been a lot of researches on regularized models, whose formulations are an objective function composed by the loss function and regularization term. These two terms are combined by the tuning parameter. It is well known that tuning parameter selection is very important. Theoretically, this parameter characterizes properties of the model solution and determines the model effect. Practically, the calculation cost and computational effect are different under different tuning parameters. As far as we known, there are three main methods to select the optimal tuning parameter, which are cross validation, information criterion and bilevel programming. For cross validation and information criterion, they all require big computational costs, causing by that fact that they need to calculate solutions under different tuning parameters. Besides that, how to appropriately choose the sequence of possible tuning parameters is an essential problem. For the purpose of reducing the computational cost of cross validation and information criterion, screening rules are proposed to eliminate inactive features in data sets and speed up the tuning parameter selection procedure. Comparing to these popular ways, transforming the tuning parameter selection problem to a bilevel programming is a more direct way. But this usually lead to a nonconvex optimization problem and still need to be explored. This paper will review existing works from tuning parameter selection methods and acceleration perspectives, respectively. Based on these, we propose the future works.

Applying the conjugate gradient method and linear projection operator, an iterative algorithm is presented to solve the least squares solution of linear matrix equation $AXB=C$ under any linear subspace. It is proved that the least squares solution, the minimum-norm least squares solution and the optimal approximation of the matrix equation $AXB=C$ can be obtained in finite iteration steps by the method without considering rounding errors. The numerical examples verify the efficiency of the algorithm. The merit of our method is that it is easy to implement in any linear subspace.

In high-dimensional data analysis, penalized quantile regression is an effective tool for variable selection and parameter estimation. In many real applications, variables are structured into groups. In order to achieve the desired effect of sparsity within and between groups, we study the sparse group lasso penalized quantile regression model that combines lasso and group Lasso. To solve computational challenges caused by non-smoothness of object function, we approximate the quantile loss function using quantile Huber function, and the quantile Huber regression with sparse group Lasso penalty (SGLQHR) is obtained. We introduce Groupwise Majorization Descent (GMD) algorithm for computing the proposed model. Numerical examples and real data analysis demonstrate the competitive performance of our algorithm.

Reconstructing clear images from blurred and noisy ones is a typical ill-posed problem. When the blurring kernel is unknown, reconstructing both the blurring kernel and the image is required. Such blind denoising and deblurring problems have attracted widespread attention in academia. Using variational methods, we established a partial differential equation model for the blind denoising and deblurring problem of remote sensing images. Then, combining alternating direction method and finite difference method, we constructed a fully discrete numerical format for solving unknown kernel functions and clear images. Numerical experiments were conducted to analyze the effect of parameters on image processing performance and to determine reasonable parameters. Finally, numerical experiments were conducted on several remote sensing images, and the results demonstrated the effectiveness of the model.

Multi-dimensional hypersingular integrals are widely used in many engineering fields such as elasticity and scattering of electromagnetic fields. In order to improve the calculation accuracy, we construct the formula of two-dimensional and three-dimensional hypersingular integrals. In this paper, the composite rectangle quadrature formula is used to approximate the part without singularity in the divided $N$ subinterval and the remaining part is solved by the analytic expression of the hypersingular integral. Based on the extrapolation, the modified composite rectangle quadrature formula of one-dimensional hypersingular integral is constructed. Finally, the modified rectangle quadrature formula is extended to the numerical quadrature of two-dimensional and three-dimensional surface hypersingular integrals. Numerical examples at the end of the paper verify the feasibility of the proposed method.

In recent years, the researches on employing artificial neural networks to solve forward and inverse problems involving partial differential equations have developed rapidly. In solving the forward problems, Penalty-Free Neural Network-2 (PFNN-2) method can accurately approximate the initial and essential boundary conditions of the problem, relax the smoothness requirement about the solution, and achieve satisfactory solution accuracy (Sheng and Yang, CiCP, 2022)^[1]. In this paper, we extend PFNN-2 to the parameter inversion problem of partial differential equation by combining its characteristics. To achieve this goal, a data-driven loss term is introduced on the basis of the original PFNN-2 loss function, and an adaptive strategy for the corresponding balance coefficient is designed. In numerical experiments, taking inversions of parameters in Burgers equation and convection-diffusion equation as examples, the proposed inversion method is tested, validating its feasibility. This study extends the application scope of the PFNN-2 method.

In this paper, we consider the convergence analysis of two accelerated iterative algorithms for solving a nonsymmetric algebraic Riccati equation arising in transport theory. This equation has two parameters $\alpha\in[0,1), c\in(0,1]$. We prove two accelerated iterative algorithms have the same convergence rates, and show that two algorithms converge linearly in $(\alpha,c)\neq(0,1)$ and sublinearly in $(\alpha,c)=(0,1)$.

In this paper, we propose a new image denoising model using the dual Lipschitz norm from optimal transport (OT) and the total variation minimization. We show the relations of this model to the $G$-TV model proposed by Yves Meyer to decomposition an image into a cartoon component, and a component representing the texture or noise. The proposed model is solved by minimizing a convex functional alternately in two variables. We design the numerical algorithm based on the Primal-Dual Hybrid Gradient (PDHG) algorithm for the Wasserstein-1 distance and the projection algorithm for the ROF model, and we establish the convergence analysis of the proposed algorithm. The existence of a minimizer of the proposed model is proved. Numerical examples demonstrate the distinct features of the proposed model compared with the traditional models such as the ROF model, and show the effectiveness of the proposed numerical method.

This paper proposes a partial linear regression model based on Fourier basis by combining nonparametric regression model and quantile regression model. We use the Fourier approximation of function model of the nonlinear function, and combined with quantile regression model estimation method of the model are given, under some basic assumptions The consistency of parameter vector and nonlinear function estimation is proved.The effectiveness of this method is demonstrated by simulation studies.At the end of the paper, the meteorological data of Beijing Capital International Airport are empirically analyzed with this model, and a new method will be proposed to accurately predict the diffusion of PM2.5 by this model.

It designs a combined high-order compact method for KdV equation in this work. This method simultaneously and compactly calculates the first-order and third-order spatial derivatives which overcomes many shortcomings of classic high-order compact methods. The KdV equation is discretized by the combined high-order compact method in space, and is approximated by the Crank-Nicolson scheme combined with extrapolation method in time. In addition, projection method is used to pull the numerical solution back to the energy-preserving manifold. Finally, some numerical experiments are conducted to verify the numerical accuracy, computational efficiency, and the property of energy-preserving.

A numerical method for solving variable-order time fractional differential equations is developed by using two-dimensional fractional-order Legendre wavelets (FOLWs). In the sense of Riemann-Liouville (R-L) variable fractional-order integral, the variable fractional-order integral formulas of FOLWs are derived by means of unit step function and regularized $\beta$ function. Based on the generalized fractional-order Taylor expansion, the error estimation of two-dimensional FOLWs expansion is studied. The variable-order time fractional differential equation is discretized into a system of algebraic equation by using the collocation method. The resulted linear and nonlinear system are solved by Gauss elimination method and Picard iterative method, respectively. The effectiveness, applicability and accuracy of the proposed method are verified by several numerical examples.

In this paper, we first provide a new convergence theorem for the greedy Kaczmarz (GK) method in reference [14]. Secondly, in order to improve the efficiency of solving consistent linear equations, a new greedy block Kaczmarz (RDBK) method based on the greedy strategy of the GK method is proposed, and the convergence theorem of the RDBK method is provided. Finally, the numerical results demonstrate that the RDBK method is significantly superior to the GK method in terms of iteration steps and computation time.

Vertex dynamics model is a mathematical model based on energy and forces, which is widely used to simulate biological processes such as cell division, cell migration, cell death and cell shape change during morphogenesis. The application of vertex dynamics model plays an important role in contribution to the understanding of biological phenomena and their underlying mechanisms. Current vertex dynamics models focus on the simulation of a tissue composed of a single type of cells. However, multiple type of tissues and cells are often involved in real biological processes. For example, the dorsal closure process during embryonic development of the model organism Drosophila melanogaster, which is accomplished by the collaboration of amnioserosa tissues consisting of flat squamous cells and epidermal tissues consisting of columnar epithelial cells. In this paper, the dorsal closure process of Drosophila embryos is successfully simulated by a vertex dynamics model, in which the above two types of cells are included, and the input data is generated by using real time-lapse images. The simulation result is very close to the experimental observation data. The construction of this model provides a new research perspective to further understand the biophysical mechanism of dorsal closure in Drosophila embryos and other related morphogenesis processes.

In this paper, a new high-order accuracy hybrid compact finite-difference scheme for solving the two-dimensional Helmholtz equation is developed using a hybrid compact finite-difference methods. To address the inefficiency of serial algorithm in solving the Helmholtz equation with large wave numbers, we propose a parallel high-order hybrid compact finite-difference algorithm based on the MPI environment on Linux cluster systems. Truncation error analysis shows that the proposed scheme has sixth-order accuracy. Numerical experimental results show that the proposed method can achieve the theoretical sixth-order accuracy in solving Helmholtz equation problems with variable wave numbers and large wave numbers. In addition, the parallel algorithm designed in this paper exhibits good parallel speedup, which can effectively improve computational efficiency.

In this present work, we propose two local energy-preserving schemes for the CamassaHolm equation, which can preserve both the local energy conservation law and the local mass conservation law, that is to say, these two schemes can accurately preserve energy and mass in any time and space regions. The local energy-preserving scheme is an extension of the global energy-preserving scheme, which eliminates the dependence on boundary conditions of the latter. Moreover, under suitable boundary conditions, such as periodic or homogeneous boundary condition, the local energy conservation law and local mass conservation law can be transformed into the corresponding global conservation laws. Finally, numerical experiments verify the splendid effect of the proposed schemes.