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This work uses multi-material topology optimization (MMTO) to maximize the average torque of a 3-phase permanent magnet synchronous machine (PMSM). Eight materials are considered in the stator: air, soft magnetic steel, three electric phases, and their three returns. To address the challenge of designing a 3-phase PMSM stator, a generalized density-based framework is used. The proposed methodology places the prescribed material candidates on the vertices of a convex polytope, interpolates material properties using Wachspress shape functions, and defines Cartesian coordinates inside polytopes as design variables. A rational function is used as penalization to ensure convergence towards meaningful structures, without the use of a filtering process. The influences of different polytopes and penalization parameters are investigated. The results indicate that a hexagonal-based diamond polytope is a better choice than the classical orthogonal domains for this MMTO problem. In addition, the proposed methodology yields high-performance designs for 3-phase PMSM stators by implementing a continuation method on the electric load angle.

Quantifying the aggregation patterns of urban population, economic activities, and land use are essential for understanding compact development, but little is known about the difference among the distribution characteristics and how the built environment influences urban aggregation. In this study, five elements are collected in Wuhan, China, namely population density, floor area ratio, business POIs, road network and built-up area as the representative of urban population, economic activities and land use. An inverse S-shape function is employed to fit the elements' macro distribution. An aggregation degree index is proposed to measure the aggregation level of urban elements. The kernel density estimation is used to identify the aggregation patterns. The spatial regression model is used to identify the built environment factors influencing the spatial distribution of urban elements. Results indicates that all urban elements decay outward from the city center in an inverse S-shape manner. The business Point-of-Interest (POI) density and population density are highly aggregated; floor area ratio and road density are moderately aggregated, whereas the built-up density is poorly aggregated. Three types of spatial aggregation patterns are identified: a point-shaped pattern, an axial pattern and a planar pattern. The spatial regression modeling shows that the built environment is associated with the distribution of the urban population, economic activities and land use. Destination accessibility factors, transit accessibility factors and land use diversity factors shape the distribution of the business POI density, floor area ratio and road density. Design factors are positively associated with population density, floor area ratio and built-up density. Future planning should consider the varying spatial concentration of urban population, economic activities and land use as well as their relationships with built environment attributes. Results of this study will provide a systematic understanding of aggregation of urban land use, population, and economic activities in megacities as well as some suggestions for planning and compact development.

In the recent decade, there has been a growing interest in using the phase-field approach to model fracture processes in various materials. Conventional phase-field implementations can simulate fracture processes using bi-linear finite element (LFE) shape functions but at the expense of a very fine mesh. In contrast, exponential finite element (EFE) shape functions can predict sharp gradients in solution variables with coarse meshes due to their exponential nature. A potential advantage lies in reducing the number of elements in the problem without losing accuracy in the solution. However, EFE shape functions do not yield a good approximation unless they are oriented relative to the expected crack propagation path. This study uses an approximate analysis using LFE shape functions to orient the EFE shape functions before the computations. Computational advantages are reported in terms of accuracy in predicted load responses and the computational times incurred.

The scaled boundary finite element method (SBFEM) is a semi-analytical approach to solving partial differential equations, in which a finite element approximation is deployed for the domain’s boundary, while analytical solutions are sought to describe the behavior in the interior of the domain. Since the inception of SBFEM, a number of different shape functions have been applied to interpolate the solution on the boundary. The overarching goal of this communication is to review the respective advantages and disadvantages of the available interpolants in the context of the SBFEM and develop recommendations regarding their application. In addition, we discuss in detail the discretization employed in the so-called diagonal SBFEM.

The hierarchical deep-learning neural network (HiDeNN) (Zhang et al. Computational Mechanics, 67:207–230) provides a systematic approach to constructing numerical approximations that can be incorporated into a wide variety of Partial differential equations (PDE) and/or Ordinary differential equations (ODE) solvers. This paper presents a framework of the nonlinear finite element based on HiDeNN approximation (nonlinear HiDeNN-FEM). This is enabled by three basic building blocks employing structured deep neural networks: (1) A partial derivative operator block that performs the differentiation of the shape functions with respect to the element coordinates, (2) An r-adaptivity block that improves the local and global convergence properties and (3) A materials derivative block that evaluates the material derivatives of the shape function. While these building blocks can be applied to any element, specific implementations are presented in 1D and 2D to illustrate the application of the deep learning neural network. Two-step optimization schemes are further developed to allow for the capabilities of r-adaptivity and easy integration with any existing FE solver. Numerical examples of 2D and 3D demonstrate that the proposed nonlinear HiDeNN-FEM with r-adaptivity provides much higher accuracy than regular FEM. It also significantly reduces element distortion and suppresses the hourglass mode.

We investigate branching processes in varying environment, for which $\\overline{f}_n \\to 1$ and $\\sum_{n=1}^\\infty (1-\\overline{f}_n)_+ = \\infty$ , $\\sum_{n=1}^\\infty (\\overline{f}_n - 1)_+ < \\infty$ , where $\\overline{f}_n$ stands for the offspring mean in generation n. Since subcritical regimes dominate, such processes die out almost surely, therefore to obtain a nontrivial limit we consider two scenarios: conditioning on nonextinction, and adding immigration. In both cases we show that the process converges in distribution without normalization to a nondegenerate compound-Poisson limit law. The proofs rely on the shape function technique, worked out by Kersting (2020).

Based on numerical shape functions and the structural stressing state theory, the mechanical properties of the curved prestressed concrete box girder (CPCBG) bridge model under different loading cases are presented. First, the generalized strain energy density (GSED) obtained from the measured strain data is used to represent the stressing state pattern of the structure; then, the stressing state of the concrete section is analyzed by plotting the strain and stress fields of the bridge model. The stressing state pattern and strain fields of the CPCBG are shown to reveal its mechanical properties. In addition, the measured concrete strain data are interpolated by the non-sample point interpolation (NPI) method. The strain and stress fields of the bridge model have been plotted to analyze the stressing state of the concrete cross-section. The internal forces in the concrete sections are calculated by using interpolated strains. Finally, the torsional effects are simulated by measuring the displacements to show the torsional behavior of the cross-section. The analysis and comparison of the internal force and strain fields reveal the common and different mechanical properties of the bridge model. The results of the analysis of the curved bridge model provide a reference for the future rational design of bridge projects.

In this paper, an implicit structural modeling method using locally defined moving least squares shape functions is proposed. The continuous bending energy is minimized to interpolate between data points and approximate geological structures. This method solves a sparse problem without relying on a complex mesh. Discontinuities such as faults and unconformities are handled with minor modifications of the method using meshless optic principles. The method is illustrated on a two-dimensional model with folds, faults and an unconformity. This model is then modified to show the ability of the method to handle sparsity, noise and different reliabilities in the data. Key parameters of the shape functions and the pertinence of the bending energy for structural modeling applications are discussed. The predefined values deduced from these studies for each parameter of the method can also be used to construct other models.

It is difficult to exactly and automatically satisfy nonseparable multipoint boundary conditions by numerical methods. With this in mind, we develop a novel algorithm to find solution for a second-order nonlinear boundary value problem (BVP), which automatically satisfies the multipoint boundary conditions prescribed. A novel concept of boundary shape function (BSF) is introduced, whose existence is proven, and it can satisfy the multipoint boundary conditions a priori. In the BSF, there exists a free function, from which we can develop an iterative algorithm by letting the BSF be the solution of the BVP and the free function be another variable. Hence, the multipoint nonlinear BVP is properly transformed to an initial value problem for the new variable, whose initial conditions are given arbitrarily. The BSF method (BSFM) can find very accurate solution through a few iterations.

Phase field models for fracture are energy-based and employ a continuous field variable, the phase field, to indicate cracks. The width of the transition zone of this field variable between damaged and intact regions is controlled by a regularization parameter. Narrow transition zones are required for a good approximation of the fracture energy which involves steep gradients of the phase field. This demands a high mesh density in finite element simulations if 4-node elements with standard bilinear shape functions are used. In order to improve the quality of the results with coarser meshes, exponential shape functions derived from the analytic solution of the 1D model are introduced for the discretization of the phase field variable. Compared to the bilinear shape functions these special shape functions allow for a better approximation of the fracture field. Unfortunately, lower-order Gauss-Legendre quadrature schemes, which are sufficiently accurate for the integration of bilinear shape functions, are not sufficient for an accurate integration of the exponential shape functions. Therefore in this work, the numerical accuracy of higher-order Gauss-Legendre formulas and a double exponential formula for numerical integration is analyzed.