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 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.

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.

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.

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.

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 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.

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.

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.

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.

This paper mainly introduces the development history of numerical computation, as well as the development and application of existing numerical computation software. In addition, the paper introduces the development history, main functionalities, development challenges, and importance of the domestically developed numerical software Baltamatica, and discusses the future development of domestic numerical software.

工业软件作为现代工业体系的核心支撑，正在有力推动制造、能源、交通等关键领域的数字化转型与智能化升级。随着工业4.0和智能制造的快速发展，工业软件在降本增效、优化设计、提质出新等方面发挥着越来越重要的作用。然而，我国在工业软件的核心技术、自主创新和产业化应用方面仍面临诸多挑战。为了促进工业软件领域的学术交流与技术突破，本专辑聚焦于工业软件的创新算法、系统开发及其在工业场景中的应用，展示了该领域的最新研究成果和前沿进展。

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.

With the evolution of design processes and the increasing integration of circuits, the scale of circuit simulation has been continuously expanding, leading to an extended simulation and verification time. The limitations of traditional hardware architectures in performance enhancement pose significant challenges for design engineers. In recent years, with the development of artificial intelligence, general-purpose graphics processing units (GPGPUs) have rapidly advanced, offering computational power and memory bandwidth far exceeding that of contemporary CPUs, thus providing new opportunities for accelerating circuit simulation. This paper will explore the mathematical models corresponding to analog and radio frequency circuit simulations, as well as the commonly used solving algorithms in the industry, analyzing the opportunities and challenges faced in heterogeneous simulation environments. By integrating advanced computing platforms, circuit simulation can achieve a more efficient solving process. Nevertheless, effectively integrating these new technologies to meet the simulation demands of complex circuits remains a key focus of current research.

Sweep surface generation is a key technique in the field of Computer-Aided Design (CAD), widely used in engineering design and other fields. To address this problem, this paper first presents the basic single-rail and dual-rail sweep algorithms for geometric splicing with a given B-spline surface, and then proposes a swept surface generation algorithm with controllable splicing errors. Subsequently, the surfaces generated by these algorithms are compared and analyzed with the results obtained by the commercial software Catia and Rhino. In the swept surface generation algorithm with controllable splicing error in this article, the boundary of the given surface is first sampled to obtain information such as derivatives and parameters, and then B-spline curve interpolation is performed on this information. Perform a sweep operation through this B-spline curve to obtain a swept surface. Finally, based on a preset threshold for splicing errors between the sweep surface and the given surface, the sampling points are adjusted to obtain a sweep surface within the acceptable error range. Experimental results show that the algorithm in this article is better than Rhino in generating surface quality, and can generate surfaces with a quality comparable to Catia with a smaller number of control points.

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.

Aircraft tires are crucial safety components during the takeoff and landing phases, which are widely recognized as the “crown jewel” of the tire industry. The design of aircraft tires is a “bottleneck” technical problem in our country. It is urgent to solve the mathematical modeling and CAE industrial software development. The aircraft tire is mainly composed of cord reinforced rubber composites, which needs to meet the complex conditions. Therefore, this paper studies the mathematical modeling of cord reinforced rubber composite for aircraft tire. According to the deformation characteristics of cord reinforced rubber composites, a modeling method of cord reinforced rubber composite is proposed to establish the embedded constraint relationship between cord and rubber structure. The finite element equations are derived for the rubber and cord structures, which are implemented in the industrial software of aircraft tire SuperTire to realize the inflation analysis. Numerical examples of aircraft tires demonstrate the modeling accuracy and solving efficiency.

Lattice Boltzmann method is an efficient algorithm for solving Boltzmann transport equation, which has been widely applied in the fields of fluid flow and heat transport of phonon. Based on our previous work of effective correction for the relaxation time of phonon, several lattice structures with different symmetries and numbers of discrete velocities are designed for the lattice Boltzmann method of phonon heat transport. The correctness of the algorithm and program is verified by a macroscale heat transport process. Then, the influences of symmetry and number of discrete velocities for different lattice structures on square-cavity heat transport at nano/microscale are studied. The results indicate that the non-physical temperature jump cannot be effectively overcome only by improving the symmetry of lattice. However, it can be effectively weakened by simultaneously improving symmetry and increasing the number of discrete velocities.