Shepard Function Research Articles

A [formula omitted] continuous multi-patch framework for adaptive isogeometric topology optimization of plate structures

This study proposes a novel computationally efficient methodology to perform topology optimization (TO) of fourth-order plate structures within the framework of multi-patch isogeometric analysis. This is realized by taking the multifold benefits of isogeometric PHT-Splines to (1) discretize the C1 continuous weak form of plate structures, (2) develop a C0 continuous density field for the material distribution in TO and inherently remove the need for filters, and (3) provide a hierarchical tree structure for the structural mesh to effortlessly implement an adaptive mesh refinement (AMR) strategy. Moreover, to ensure continuity between isogeometric patches, we adopt a strong C1 coupling between the boundaries. This is established by constructing new basis functions, defined as a linear combination of existing C0 functions at the patch interfaces. The density field in TO is further enhanced with a first-neighbourhood smoothening algorithm based on the Shepard function to generate printable topologies and alleviate the post-processing stages after optimization. An element-centre density, based on the control point densities of the isogeometric mesh, is used as the marking scheme for the AMR to determine the subdomains to be refined. Utilizing the Geometry Independent Field approximaTion, the design and adaptive analysis-optimization stages were independently discretized respectively through NURBS and PHT-Splines, allowing easy transfer of multi-patch geometries from industry-standard packages. Multiple numerical examples illustrate the stability of the multi-patch algorithm in optimizing the geometries effectively. The results also show considerable advantages in terms of solution accuracy such as precise field, smooth topology and computational efficiency.

A continuous field adaptive mesh refinement algorithm for isogeometric topology optimization using PHT-Splines

This study proposes a continuous density-field based isogeometric topology optimization (ITO) using Polynomial splines over Hierarchical T-meshes (PHT-Splines). Taking the benefit of isogeometric methods, the design density values for the mesh are defined using spline basis functions as a continuous density function (CDF) through the control points. The material density at the control points will then appear as a facade over the domain, which is further employed to generate a very smooth topology and also to propose a marking criterion for an adaptive mesh refinement (AMR) strategy during the optimization process. During this adaptive meshing, the isogeometric domain mesh is refined during each adaptive step, properly tracking the density variation over the elements and sharpening the boundaries of the optimized topologies. By utilizing the rationale of the Geometry Independent Field approximaTion (GIFT) framework, any complex multi-patch geometry constructed in an industry-standard CAD package using Non-Uniform Rational B-Splines (NURBS) may be imported and discretized using PHT-Splines, including higher dimensions. This helps to maintain the geometrical accuracy through an initial coarse mesh and computational accuracy of the optimization iterations through the local refinement feature of the PHT-Splines. An adaptive first-neighbourhood smoothening algorithm based on the Shepard function is also proposed to obtain manufacturable topologies and to circumvent post-processing stages. The proposed method was observed to provide remarkably smooth topologies with an initial coarse mesh, thereby obtaining a computationally efficient domain to be solved. Compared to existing element density based methods, the proposed method demonstrates a substantial minimization in Degree’s-of-Freedom (DoF) requirement to arrive at clear and realistic solutions. Also, when compared to globally refined solutions, a 50%–90% reduction in DoF is achieved using the adaptive refinement strategy.

Lattice Discrete Particle Modeling of Size Effect in Slab Scratch Tests

AbstractFor a long time, geomechanicians have used scratch tests to characterize the compressive behavior and hardness of rocks. In recent years, this test has regained popularity in the field of mechanics, especially after a series of publications that highlighted the potential capability of the scratch test to determine the fracture properties of quasi-brittle materials. However, the complex failure mechanisms observed experimentally in scratch tests led to scientific debates and, in particular, raised the question of the size effect. This article intends to provide a better understanding of the problem by using numerical tools and fracture mechanics considerations. To narrow the investigation area, this study focuses on slab scratch tests of quasi-brittle materials and adopts two different numerical methods: (i) the lattice discrete particle model (LDPM) that includes constitutive laws for cohesive fracturing, frictional shearing, and nonlinear compressive behavior, and (ii) the meshless method based on Shepard function and partition of unity (MSPU) implementing linear elastic fracture mechanics (LEFM). The numerical results are further analyzed through Bažant’s size effect law (SEL) with an appropriate mixed-mode fracture criterion. Fracture properties are then calculated and compared to the results of typical notched three-point bending tests. The results show that mixed-mode fracture considerations are of paramount importance in analyzing the fracture process and size effect of scratch tests.

Automatic digital-numerical integrated analysis by the image-based meshless method

The paper takes advantage of the digital images exported from the digital visualization models of geotechnical engineering and the meshless method based on Shepard function and Partition of Unity (MSPU) to develop a new solution for the automatic digital-numerical integrated analysis. The proposed method fully utilizes the information of pixels to substitute geometric models with digital images, which avoids complex and time-consuming computational geometry in conventional pre-processing procedure. A regular and structured mesh is adopted to cover the digital image and the central points of the rectangular elements are taken as the interpolation nodes to discrete the analysis domain. The paper introduces in detail the formulation of the corresponding MSPU interpolation, the treatments of interfaces between materials and the imposition of the displacement boundary conditions. The method possesses the merits of concise formulation, simple numerical implementation and high automation. The proposed method can reduce the high requirements to computational mechanics and professional numerical modeling skills for geotechnical engineers, and provide a new convenient means for automatic numerical analysis based on digital information model. Numerical examples including representative elastic problems and slope stability analysis indicate the high accuracy and wide prospect of the proposed method.

Isogeometric topology optimization for rational design of ultra-lightweight architected materials

Architected materials with the desirable properties, such as the lightweight and the superior stiffness, have accepted considerable attention in recent years. This paper aims to study the design of architected materials but with the ultra-lightweight using an effective and efficient Isogeometric Topology Optimization (ITO) method. An enough smooth and continuous Density Distribution Function (DDF) is constructed using the Shepard function and NURBS basis functions, to describe the topology of the micro-architecture. Later, the homogenization with a simple periodic boundary formulation is numerically implemented by isogeometric analysis (IGA) to predict effective macroscopic properties using the micro information. An ITO formulation with the specific objective function is developed to rationally design 2D and 3D architected materials with the extreme elastic properties using the DDF and IGA-based homogenization, in which the volume fraction needs to be very low to ensure the optimized micro-architectures with the ultra-lightweight. NURBS basis functions offer a unified formula for the structural geometry, the solution space and the micro topology in architected materials. Finally, several numerical examples are provided to display the effectiveness of the ITO method for the ultra-lightweight architected materials, particularly for 3D scenario. A series of novel and interesting 3D ultra-lightweight architected materials are found, which are also prototyped using the Selective Laser Sintering (SLS) technique.

Isogeometric topology optimization for continuum structures using density distribution function

SummaryThis paper will propose a more effective and efficient topology optimization method based on isogeometric analysis, termed as isogeometric topology optimization (ITO), for continuum structures using an enhanced density distribution function (DDF). The construction of the DDF involves two steps. (1) Smoothness: the Shepard function is firstly utilized to improve the overall smoothness of nodal densities. Each nodal density is assigned to a control point of the geometry. (2) Continuity: the high‐order NURBS basis functions are linearly combined with the smoothed nodal densities to construct the DDF for the design domain. The nonnegativity, partition of unity, and restricted bounds [0, 1] of both the Shepard function and NURBS basis functions can guarantee the physical meaning of material densities in the design. A topology optimization formulation to minimize the structural mean compliance is developed based on the DDF and isogeometric analysis to solve structural responses. An integration of the geometry parameterization and numerical analysis can offer the unique benefits for the optimization. Several 2D and 3D numerical examples are performed to demonstrate the effectiveness and efficiency of the proposed ITO method, and the optimized 3D designs are prototyped using the Selective Laser Sintering technique.

Topology optimization for auxetic metamaterials based on isogeometric analysis

In this paper, an effective and efficient topology optimization method, termed as Isogeometric Topology Optimization (ITO), is proposed for systematic design of both 2D and 3D auxetic metamaterials based on isogeometric analysis (IGA). Firstly, a density distribution function (DDF) with the desired smoothness and continuity, to represent the topological changes of structures, is constructed using the Shepard function and non-uniform rational B-splines (NURBS) basis functions. Secondly, an energy-based homogenization method (EBHM) to evaluate material effective properties is numerically implemented by IGA, with the imposing of the periodic boundary formulation on material microstructure. Thirdly, a topology optimization formulation for 2D and 3D auxetic metamaterials is developed based on the DDF, where the objective function is defined as a combination of the homogenized elastic tensor and the IGA is applied to solve the structural responses. A relaxed optimality criteria (OC) method is used to update the design variables, due to the non-monotonic property of the problem. Finally, several numerical examples are used to demonstrate the effectiveness and efficiency of the proposed method. A series of auxetic microstructures with different deformation mechanisms (e.g. the re-entrant and chiral) can be obtained. The auxetic behavior of material microstructures are numerically validated using ANSYS, and the optimized designs are prototyped using the Selective Laser Sintering (SLS) technique.

A mixed cover meshless method for elasticity and fracture problems

A mixed cover meshless method (MCMM) is developed to solve elasticity and fracture problems. In this technique, an arbitrary computational geometry is discretized using regular square cells, and meshless approximation functions are separately defined at the interior and boundary square cells using the concept of independent nodal covers and overlapping nodal covers, respectively. For the fracture analysis, a set of triangular independent nodal covers around a crack tip is employed, and the virtual crack closure technique (VCCT) is used to calculate the crack-tip stress intensity factors (SIFs). The overlapping nodal covers and independent nodal covers can be freely selected and converted as required during the simulation process of crack growth, and the square cells near geometry boundaries, such as material discontinuities, crack lines, or crack tips, can be further subdivided by quadtree decomposition to perform h-adaptivity analysis and to achieve the desired solution accuracy. The MCMM gets rid of the need for generating conforming meshes in the finite element method (FEM), possesses the merits of a concise formulation of interpolation functions, simple numerical implementation and convenient simulation of crack growth along arbitrary directions, and improves the computational efficiency compared to classic meshless methods such as the element-free Galerkin method (EFGM) and the meshless method based on Shepard function and partition of unity (MSPU). Several representative elasticity and fracture examples demonstrate the convergence, accuracy, and robustness of the present method.

Geometrically nonlinear analysis of thick orthotropic plates with various geometries using simple HP-cloud method

Purpose – The purpose of this paper is to present a simple HP-cloud method as an accurate meshless method for the geometrically nonlinear analysis of thick orthotropic plates of general shape. This method is used to investigate the effects of thickness, geometry of various shapes, boundary conditions and material properties on the large deformation analysis of Mindlin plates. Design/methodology/approach – Nonlinear analysis of plates based on Mindlin theory is presented. The equations are derived by the Von-Karman assumption and total Lagrangian formulations. Newton-Raphson method is applied to achieve linear equations from nonlinear equations. Simple HP-cloud method is used for the construction of the shape functions based on Kronecker-δ properties, so the essential boundary conditions can be enforced directly. Shepard function is utilized for a partition of unity and complete polynomial is used as an enrichment function. Findings – The suitability and efficiency of the simple HP-cloud method for the geometrically nonlinear analysis of thin and moderately thick plates is studied for the first time. Large displacement analysis of various shapes of plates, rectangular, skew, trapezoidal, circular, hexagonal and triangular with different boundary conditions subjected to distributed loading are considered. Originality/value – This paper shows that the simple HP-cloud method is well suited for the large deformation analysis of Mindlin plates with various geometries, because it uses a set of a few arbitrary nodes placed in a plate of general shape. Moreover the convergence rate of the proposed method is high and the cost of solving equations is low.

Meshless method based on Shepard function and partition of unity for two-dimensional crack problems

A new Meshless method based on Shepard function and Partition of Unity (MSPU) is proposed for calculating crack SIFs (Stress Intensity Factors) and simulating crack propagation. Link elements are employed to connect the adjacent block elements inside and around the circle of a crack tip, and to solve the challenging problem of imposing essential boundary conditions for meshless methods. The proposed MSPU possesses the merits of concise interpolation formulation and simple numerical implementation, and shows many advantages over existing meshless methods. In this work, the virtual crack closure technique (VCCT) is used to capture the crack tip SIFs, and the crack propagation is determined based on the maximum circumferential stress criterion. Numerical examples of representative cracking problems indicate that the MSPU is of high accuracy, good stability and sufficient convergence rate in fracture analyses, and has a wide application prospective.

Topology optimization of compliant mechanisms using element-free Galerkin method

This paper will propose a topology optimization approach for the design of large displacement compliant mechanisms with geometrical non-linearity by using the element-free Galerkin (EFG) method. In this method, the Shepard function is applied to construct a physically meaningful density approximant, to account for its non-negative and range-bounded property. Firstly, in terms of the original nodal density field, the Shepard function method functionally similar to a density filter is used to generate a non-local nodal density field with enriched smoothness over the design domain. The density of any node can be evaluated according to the nodal density variables located inside the influence domain of the interested node. Secondly, in the numerical implementation the Shepard function method is again employed to construct a point-wise density interpolant. Gauss quadrature is used to calculate the integration of background cells numerically, and the artificial densities over all Gauss points can be determined by the surrounding nodal densities within the influence domain of the concerned computational point. Finally, the moving least squares (MLS) method is applied to construct the shape functions using the weight functions with compact support for assembling the meshless approximations of state equations. Since MLS shape functions are lack of the Kronecker delta function property, the penalty method is applied to enforce the essential boundary conditions. A typical large-deformation compliant mechanism is used as the numerical example to demonstrate the effectiveness of the proposed method.

Topology optimization with meshless density variable approximations and BESO method

An improved meshless density variable approximation is incorporated into the BESO method for topology optimization of continuum structures in this paper. The essential boundary condition is enforced by using the compactly supported radial basis function (CSRBF). The Shepard function is used to create a physically meaningful dual-level density approximation. Numerical examples show that the proposed method is feasible and fidelity for the topology optimization of continuum structures. The common numerical instabilities of the BESO method do not exist in the final results.

Topology optimization of structures using meshless density variable approximants

SUMMARYThis paper proposes a new structural topology optimization method using a dual‐level point‐wise density approximant and the meshless Galerkin weak‐forms, totally based on a set of arbitrarily scattered field nodes to discretize the design domain. The moving least squares (MLS) method is used to construct shape functions with compactly supported weight functions, to achieve meshless approximations of system state equations. The MLS shape function with the zero‐order consistency will degenerate to the well‐known ‘Shepard function’, while the MLS shape function with the first‐order consistency refers to the widely studied ‘MLS shape function’. The Shepard function is then applied to construct a physically meaningful dual‐level density approximant, because of its non‐negative and range‐restricted properties. First, in terms of the original set of nodal density variables, this study develops a nonlocal nodal density approximant with enhanced smoothness by incorporating the Shepard function into the problem formulation. The density at any node can be evaluated according to the density variables located inside the influence domain of the current node. Second, in the numerical implementation, we present a point‐wise density interpolant via the Shepard function method. The density of any computational point is determined by the surrounding nodal densities within the influence domain of the concerned point. According to a set of generic design variables scattered at field nodes, an alternative solid isotropic material with penalization model is thus established through the proposed dual‐level density approximant. The Lagrangian multiplier method is included to enforce the essential boundary conditions because of the lack of the Kronecker delta function property of MLS meshless shape functions. Two benchmark numerical examples are employed to demonstrate the effectiveness of the proposed method, in particular its applicability in eliminating numerical instabilities. Copyright © 2012 John Wiley & Sons, Ltd.

A critical assessment of the truly Meshless Local Petrov-Galerkin (MLPG), and Local Boundary Integral Equation (LBIE) methods

The essential features of the Meshless Local Petrov-Galerkin (MLPG) method, and of the Local Boundary Integral Equation (LBIE) method, are critically examined from the points of view of a non-element interpolation of the field variables, and of the meshless numerical integration of the weak form to generate the stiffness matrix. As truly meshless methods, the MLPG and the LBIE methods hold a great promise in computational mechanics, because these methods do not require a mesh, either to construct the shape functions, or to integrate the Petrov-Galerkin weak form. The characteristics of various meshless interpolations, such as the moving least square, Shepard function, and partition of unity, as candidates for trial and test functions are investigated, and the advantages and disadvantages are pointed out. Emphasis is placed on the characteristics of the global forms of the nodal trial and test functions, which are non-zero only over local sub-domains ΩtrJ and ΩteI, respectively. These nodal trial and test functions are centered at the nodes J and I (which are the centers of the domains ΩtrJ and ΩteI), respectively, and, in general, vanish at the boundaries ∂ΩtrJ and ∂ΩteI of ΩtrJ and ΩteI, respectively. The local domains ΩtrJ and ΩteI can be of arbitrary shapes, such as spheres, rectangular parallelopipeds, and ellipsoids, in 3-Dimensional geometries. The sizes of ΩtrJ and ΩteI can be arbitrary, different from each other, and different for each J, and I, in general. It is shown that the LBIE is but a special form of the MLPG, if the nodal test functions are specifically chosen so as to be the modified fundamental solutions to the differential equations in ΩteI, and to vanish at the boundary ∂ΩteI. The difficulty in the numerical integration of the weak form, to generate the stiffness matrix, is discussed, and a new integration method is proposed. In this new method, the Ith row in the stiffness matrix is generated by integrating over the fixed sub-domain ΩteI (which is the support for the test function centered at node I); or, alternatively the entry KIJ in the global stiffness matrix is generated by integrating over the intersections of the sub-domain ΩtrJ (which is the sub-domain, with node J as its center, and over which the trial function is non-zero), with ΩteI (which is the sub-domain centered at node I over which the test function is non-zero). The generality of the MLPG method is emphasized, and it is pointed that the MLPG can also be the basis of a Galerkin method that leads to a symmetric stiffness matrix. This paper also points out a new but elementary method, to satisfy the essential boundary conditions exactly, in the MLPG method, while using meshless interpolations of the MLS type. This paper presents a critical appraisal of the basic frameworks of the truly meshless MLPG/LBIE methods, and the numerical examples show that the MLPG approach gives good results. It now apears that the MLPG method may replace the well-known Galerkin finite element method (GFEM) as a general tool for numerical modeling, in the not too distant a future.