Fictitious Density Research Articles

Analysis of Numerical Micromodulus Coupled with Influence Function for Brittle Materials via Bond-Based Peridynamics

In this paper, the numerical micromodulus is derived for the plane stress problem to develop a new insight into the application of bond-based peridynamics. Considering the nonlocal property of peridynamics, the numerical micromodulus coupled with influence function provides a reasonable description of the long-range force effect. Through several numerical applications, the effectiveness of the numerical modulus coupled with various influence functions to simulate deformation and failure is analyzed. In addition, a load increment algorithm based on fictitious density is developed specifically for quasi-static problems. It is indicated that the introduction of the influence function can enhance the accuracy in deformation and failure simulation, which is valuable for the advancement and application of numerical micromoduli. Through a comprehensive trade-off between simulation accuracy and stability, the numerical micromodulus coupled with the exponential influence function proves to be the more effective option for brittle material.

Direction-oriented stress-constrained topology optimization of orthotropic materials

Efficient optimization of topology and raster angle has shown unprecedented enhancements in the mechanical properties of 3D printed materials. Topology optimization helps reduce the waste of raw material in the fabrication of 3D printed parts, thus decreasing production costs associated with manufacturing lighter structures. Fiber orientation plays an important role in increasing the stiffness of a structure. This paper develops and tests a new method for handling stress constraints in topology and fiber orientation optimization of 3D printed orthotropic structures. The stress constraints are coupled with an objective function that maximizes stiffness. This is accomplished by using the modified solid isotropic material with penalization method with the method of moving asymptotes as the mathematical optimizer. Each element has a fictitious density and an angle as the main design variables. To reduce the number of stress constraints and thus the computational cost, a new clustering strategy is employed in which the highest stresses in the principal material coordinates are grouped separately into two clusters using an adjusted P-norm. A detailed description of the formulation and sensitivity analysis is discussed. While we present an analysis of 2D structures in the numerical examples section, the method can also be used for 3D structures, as the formulation is generic. Our results show that this method can produce efficient structures suitable for 3D printing while thresholding the stresses.

On some applications of Generalized Geometric Projection to optimal 3D printing

In recent years, Topology Optimization (TO) gained interest in the scientific community. It assists in finding the best arrangement of material in a design volume. The classical approach named ”Solid Isotropic Material with Penalization” (SIMP) associates a fictitious density to each finite element in the domain. While SIMP is described as an implicit approach which can lead to problems with dimensionality of variables, explicit methods adopt a geometric projection of simple elements (eg.: bars) to reduce the number of design variables. This simplifies the geometric interpretation of the optimal architecture. The major explicit methods were recently unified into a general framework, Generalized Geometric Projection (GGP).Currently it is quite challenging to take into account manufacturing constraints in the topology optimization design phase. Therefore this paper presents an application of the GGP Method to the design of products made by Additive Layer Manufacturing (ALM). Every printed layer constitutes a geometric element, involving design variables relative to position and width. Specific constraints of ALM, including bridge length and overhang angle, can be easily monitored by exploiting the geometric features of the combined elements.Examples in two dimensions will be reported, analyzing two academic benchmark problems. A comparison to other proven techniques is also detailed. An mean difference of 7.7% is observed for solutions with only overhang angle constraint, while a mean difference of 11% is observed for solutions with overhang angle and bridge length constraint. The presented work integrates design and manufacturing, directly identifying the path of the printed layers.

Indirect boundary element method for modelling 2‐D poroelastic wave diffraction by cavities and cracks in half space

AbstractAn indirect boundary element method (IBEM) is developed to solve two‐dimensional (2‐D) wave diffraction by cavities and cracks in a fluid‐saturated poroelastic half space. The dynamic Green's functions of forces and fluid source in a poroelastic full‐space, rather than their half‐space counterparts, are utilized, because of their closed‐form expressions. The diffracted wave field is constructed by distributing forces and sources in fictitious densities over the boundaries, based on the single‐layer potential theory. The densities are determined by matching the approximate solution with the boundary conditions. The nonuniqueness of integral equation caused by the degenerated geometry of a crack is treated by a zoned approach. The numerical accuracy and stability of the IBEM are verified by comparing with available results in the literature. Problems of wave scattering by cavity and flat crack in saturated half plane are solved. The displacement amplification effect on the free surface is investigated. The numerical solutions show that the response is strongly dependent on factors such as wave frequency, incident angle, the length and depth of the crack, and material properties.

Topology optimization of load-bearing capacity

The present work addresses the problem of maximizing a structure load-bearing capacity subject to given material strength properties and a material volume constraint. This problem can be viewed as an extension to limit analysis problems which consist in finding the maximum load capacity for a fixed geometry. We show that it is also closely linked to the problem of minimizing the total volume under the constraint of carrying a fixed loading. Formulating these topology optimization problems using a continuous field representing a fictitious material density yields convex optimization problems which can be solved efficiently using state-of-the-art solvers used for limit analysis problems. We further analyze these problems by discussing the choice of the material strength criterion, especially when considering materials with asymmetric tensile/compressive strengths. In particular, we advocate the use of a L1-Rankine criterion which tends to promote uniaxial stress fields as in truss-like structures. We show that the considered problem is equivalent to a constrained Michell truss problem. Finally, following the idea of the SIMP method, the obtained continuous topology is post-processed by an iterative procedure penalizing intermediate densities. Benchmark examples are first considered to illustrate the method overall efficiency while final examples focus more particularly on no-tension materials, illustrating how the method is able to reproduce known structural patterns of masonry-like structures. This paper is accompanied by a Python package based on the FEniCS finite-element software library.

Hole seeding in level set topology optimization via density fields

Two approaches that use a density field for seeding holes in level set topology optimization are proposed. In these approaches, the level set field describes the material-void interface while the density field describes the material distribution within the material phase. Both fields are optimized simultaneously by coupling them through either a single abstract design variable field or a penalty term introduced into the objective function. These approaches eliminate drawbacks of level set topology optimization methods that rely on seeding the initial design domain with a large number of holes. Instead, the proposed approaches insert holes during the optimization process where beneficial. The dependency of the optimization results on the initial hole pattern is reduced, and the computational costs are lowered by keeping the number of elements intersected by the material interface at a minimum. In comparison with level set methods that use topological derivatives to seed small holes at distinct steps in the optimization process, the proposed approaches introduce holes continuously during the optimization process, with the hole size and shape being optimized for the particular design problem. The proposed approaches are studied using the extended finite element method for spatial discretization, and the solid isotropic material with penalization for material interpolation using fictitious densities. Their robustness with respect to algorithmic parameters, dependency on the density penalization, and performance are examined through 2D and 3D benchmark linear elastic numerical examples, and a geometrically complex mass minimization with stress constraint design problem.

N‐centered ensemble density‐functional theory for open systems

AbstractTwo (so‐called left and right) variants of N‐centered ensemble density‐functional theory (DFT) are presented. Unlike the original formulation of the theory, these variants allow for the description of systems with a fractional electron number. While conventional DFT for open systems uses only the true electron density as basic variable, left/right N‐centered ensemble DFT relies instead on (a) a fictitious ensemble density that integrates to a central (integral) number N of electrons, and (b) a grand canonical ensemble weight α which is equal to the deviation of the true electron number from N. Within such a formalism, the infamous derivative discontinuity that appears when crossing an integral number of electrons is described exactly through the dependence in α of the left and right N‐centered ensemble Hartree‐exchange‐correlation density functionals. Incorporating N‐centered ensembles into existing density‐functional embedding theories is expected to pave the way toward the in‐principle‐exact description of an open fragment by means of a pure‐state N‐electron many‐body wavefunction. Work is currently in progress in this direction.

A modified axisymmetric ordinary state-based peridynamics with shear deformation for elastic and fracture problems in brittle solids

Although axisymmetric problems are neither the plane stress nor plane strain problems, they can be solved in the two-dimensional (2-D) domain for convenience. In this paper, a modified axisymmetric ordinary state-based peridynamic model considering the shear deformation for linear elastic solids is proposed. By subtracting the rigid body rotation part from the total deformation, the shear deformation of a bond is considered. The bond force density vector is derived indirectly by equalizing the PD energy density and the classical strain energy density. The fictitious density in adaptive dynamic relaxation (ADR) method is derived. Two new damage criterions based on the maximum stretch/deviatoric strain are proposed. Several axisymmetrically three-dimensional (3-D) numerical examples are performed in the computational domains, where the present numerical results are compared with the existing analytical solutions, the classical finite element numerical results and axisymmetric ordinary state-based peridynamic results. The numerical results demonstrate this proposed method can simulate the initiation and propagation of cracks very well under the axisymmetrical conditions, and it has more accuracy than axisymmetric ordinary state-based peridynamic model.

Vanishing skyrmion Hall effect at the angular momentum compensation temperature of a ferrimagnet.

In the presence of a magnetic field, the flow of charged particles in a conductor is deflected from the direction of the applied force, which gives rise to the ordinary Hall effect. Analogously, moving skyrmions with non-zero topological charges and finite fictitious magnetic fields exhibit the skyrmion Hall effect, which is detrimental for applications such as skyrmion racetrack memory. It was predicted that the skyrmion Hall effect vanishes for antiferromagnetic skyrmions because their fictitious magnetic field, proportional to net spin density, is zero. Here we investigate the current-driven transverse elongation of pinned ferrimagnetic bubbles. We estimate the skyrmion Hall effect from the angle between the current and the bubble elongation directions. The angle and, hence, the skyrmion Hall effect vanishes at the angular momentum compensation temperature where the net spin density vanishes. Furthermore, our study establishes a direct connection between the fictitious magnetic field and the spin density.

Topology optimization of conductors in electrical circuit

This study proposes a topology optimization method for realizing a free-form design of conductors in electrical circuits. Conductors in a circuit must connect components, such as voltage sources, resistors, capacitors, and inductors, according to the given circuit diagram. The shape of conductors has a strong effect on the high-frequency performance of a circuit due to parasitic circuit elements such as parasitic inductance and capacitance. In this study, we apply topology optimization to the design of such conductors to minimize parasitic effects with maximum flexibility of shape manipulation. However, when the distribution of conductors is repeatedly updated in topology optimization, disconnections and connections of conductors that cause open and short circuits, respectively, may occur. To prevent this, a method that uses fictitious electric current and electric field calculations is proposed. Disallowed disconnections are prevented by limiting the maximum value of the fictitious current density in conductors where a current is induced. This concept is based on the fact that an electric current becomes concentrated in a thin conductor before disconnection occurs. Disallowed connections are prevented by limiting the maximum value of the fictitious electric field strength around conductors where a voltage is applied. This is based on the fact that the electric field in a parallel plate capacitor is inversely proportional to the distance between the plates. These limitations are aggregated as a single constraint using the Kreisselmeier-Steinhauser function in the formulation of optimization problems. This constraint prevents only disallowed disconnections and connections, but does not prevent allowed topology changes. The effectiveness of the constraint is confirmed using simple examples, and an actual design problem involving conductors in electromagnetic interference filters is used to verify that the proposed constraint can be utilized for conductor optimization.

An axisymmetric ordinary state-based peridynamic model for linear elastic solids

In this study, a new axisymmetric ordinary state-based peridynamic (PD) model for axisymmetric problems of linear elastic solids is presented. A fracture criterion based on the PD bond energy density is proposed. Adaptive dynamic relaxation (ADR) method is adopted to obtain equilibrium solutions, and a viable fictitious density of the model is derived and proven to be valid for implementation in the ADR method. Performance and validity of the proposed axisymmetric PD model are demonstrated by three kinds of numerical problems, i.e., compression tests, pull-out deformation, and indentation fracture. In the compression tests with focus on constant-strain deformations, the displacements predicted by the present model are compared with classical analytical solutions, and the results show good agreements. Both m- convergence and δ-convergence behaviors under four influence functions are investigated and discussed based on a thorough error analysis in different compression cases. The recovery of the Poisson’s ratios in the model is tested in detail as well. The capability of capturing general non-uniform axisymmetric deformation by the proposed model is verified in the pull-out analysis. The equilibrium displacement fields predicted by the present model agree very well with those by the finite element method. The peridynamic evaluation of strains and stresses also show good match with the finite element ones. The proposed fracture criterion is validated by the third example of indentation cracking which is compared with the available experimental data. The model developed can effectively be used to analyze axisymmetric problems of linear elastic solids in the framework of the ordinary state-based peridynamics.

An isogeometric approach of two dimensional acoustic design sensitivity analysis and topology optimization analysis for absorbing material distribution

A study of structural shape optimization and absorbing material distribution optimization of noise barrier structures based on the recently proposed isogeometric analysis method with exact geometric definitions is presented. The acoustic scattering is approximated using the basis functions that represent the geometry. A fast multipole method is applied to accelerate the solution of the boundary element method (BEM). The Burton–Miller formulation is used to overcome the fictitious frequency problem when a single Helmholtz boundary integral equation is used for the exterior boundary–value problem. The strongly singular integrals in the Burton–Miller formulation using the isogeometric BEM are evaluated explicitly and directly, particularly for the sensitivity formulation with a hyper-singular integral. The optimality criteria method is used for two types of optimization analyses, shape optimization and material distribution topology optimization. For the shape optimization, the design variables can be set to the locations of the control points because the control points determine the shape of structure. For the second optimization, a new material interpolation scheme for acoustic problems based on the solid isotropic material with penalization (SIMP) method is given, where the interpolation variable is not the real structural density used in a conventional SIMP, but a fictitious material density that determines the normalized surface admittance. Several examples are presented to demonstrate the validity and efficiency of the proposed algorithm.

Study on the Optimization of the Distribution of Absorbing Material on a Noise Barrier

Noise barriers have been widely used to decrease the transportation noise. How to utilize limited structure and material to achieve the environmental requirements for noise control is an important research topic. This research combines the use of boundary element method and SIMP to generate an acoustic topology optimization method and applies this optimization method to optimization distribution of absorbing material installed on the noise barrier’s edges. Optimality criteria method is used to update the design variables and look for the final optimal solution. A new material interpolation scheme for acoustic problems based on SIMP is given, where the interpolation variable is not real structural density used in conventional SIMP, but fictitious material density deciding the normalized surface admittance. It is noteworthy that gray elements exist in the acoustic optimization analysis based on SIMP. However, a modified method based on a smoothed Heaviside function is applied to eliminate the gray elements. In order to demonstrate the validity and efficiency of the proposed algorithm in this paper, vertical barrier and T-shaped barrier with two wells are used for the numerical analysis, respectively.

Gradient-based multi-component topology optimization for stamped sheet metal assemblies (MTO-S)

This paper presents a continuous relaxation of our previous work on the multi-component topology optimization, which enables the use of efficient gradient-based optimization to attain the simultaneous optimization of a base structural topology and its decomposition. In addition to the fictitious material density used in the classic SIMP method, new design variables specifying fractional membership to each component are introduced, in order to relax the process of decomposing a candidate structural topology into a discrete number of components. The concept of joint stiffness is also relaxed using the new component membership variables. Considering stamped sheet metal assemblies jointed by the resistance spot-welding, the stamping die cost consists of die-set material cost and die machining cost. The former is modeled as the minimum bounding box area enclosing each component, approximated by a product of weighted variances in major and minor directions obtained by a weighted principal component analysis. The latter is modeled as an empirical shape complexity index similar to the one available in the literature. Examples on compliance minimization of multi-component sheet metal assemblies are presented, where the proposed continuous relaxation formulation solved by a gradient-based optimization algorithm generated comparable results with dramatically improved computational efficiency over our previous results by discrete formulations solved by the genetic algorithm.

Hybrid optimization of a vibration isolation system considering layout of structure and locations of components

A vibration isolation system is designed using novel hybrid optimization techniques, where locations of machines, locations of isolators and layout of supporting structure are all taken as design variables. Instead of conventional parametric optimization model, the 0-1 programming model is established to optimize the locations of machines and isolators so that the time-consuming remeshing procedure and the complicated sensitivity analysis with respect to position parameters can be circumvented. The 0-1 sequence for position design variables is treated as binary bits so as to reduce the actual number of design variables to a great extent. This way the 0-1 programming can be solved in a quite efficient manner using a special version of genetic algorithm(GA) that has been published by the authors. The layout of supporting structure is optimized using SIMP based topology optimization method, where the fictitious elemental densities are taken as design variables ranging from 0 to 1. Influence of different design variables is firstly investigated by numerical examples. Then a hybrid multilevel optimization method is proposed and implemented to simultaneously take all design variables into account.

Deflection of Laminated Composite Plates Using Dynamic Relaxation Method

First – order orthotropic shear deformation equations for the nonlinearly elastic bending response of rectangular plates are introduced. Their solution using a computer program based on finite differences implementation of the Dynamic Relaxation (DR) method is outlined. The convergence and accuracy of the DR solutions for elastic large deflection response of isotropic, orthotropic, and laminated plates are established by comparison with various exact and approximate solutions. The present Dynamic Relaxation method (DR) coupled with finite differences method shows a fairly good agreement with other analytical and numerical methods used in the verification scheme. It was found that: The convergence and accuracy of the DR solution are dependent on several factors including boundary conditions, mesh size, and type, fictitious densities, damping coefficients, time increment and applied load. Also, the DR large deflection program using uniform finite differences meshes can be employed in the analysis of different thicknesses for isotropic, orthotropic, or laminated plates under uniform loads. All the comparison results for simply supported (SS4) edge conditions showed that deflection is almost dependent on the direction of the applied load or the arrangement of the layers.

Optimization of Ply Drop Locations in Variable-Stiffness Composites

A method for optimizing ply drop locations in composite laminates is developed using ideas from topology optimization. The design is parametrized using a fiber angle and fictitious density distribution for each ply. The solution proceeds using a successive conservative convex approximations strategy. Starting from a feasible point, the algorithm converges to a local minimum. The structural responses are approximated separately in terms of the angle and fictitious density distributions of each ply. Explicit penalization of intermediate densities is used to force the densities to either one for ply coverage or zero for no coverage. Ply drop locations are identified as the boundary between regions having densities near one and those near zero. The ply drop optimization is combined with fiber angle optimization by alternating the optimization between the corresponding sets of variables. Optimized variable-stiffness variable-thickness composite laminates, with a prespecified ply drop order, are obtained. Initial results show that large improvements in the buckling loads of flat plates can be obtained by combining ply drop and fiber angle optimizations.

Derivative discontinuity and exchange-correlation potential of meta-GGAs in density-functional theory.

We investigate fundamental properties of meta-generalized-gradient approximations (meta-GGAs) to the exchange-correlation energy functional, which have an implicit density dependence via the Kohn-Sham kinetic-energy density. To this purpose, we construct the most simple meta-GGA by expressing the local exchange-correlation energy per particle as a function of a fictitious density, which is obtained by inverting the Thomas-Fermi kinetic-energy functional. This simple functional considerably improves the total energy of atoms as compared to the standard local density approximation. The corresponding exchange-correlation potentials are then determined exactly through a solution of the optimized effective potential equation. These potentials support an additional bound state and exhibit a derivative discontinuity at integer particle numbers. We further demonstrate that through the kinetic-energy density any meta-GGA incorporates a derivative discontinuity. However, we also find that for commonly used meta-GGAs the discontinuity is largely underestimated and in some cases even negative.

Verification of Dynamic Relaxation (DR) Method in Isotropic , Orthotropic and Laminated Plates using Small Deflection Theory

Dynamic Relaxation (DR) method is presented for the analysis of geometrically linear laterally loaded, rectangular laminated plates. The analysis uses the Mindlin plate theory which accounts for transverse shear deformations. A computer program has been compiled. The convergence and accuracy of the DR solutions of isotropic, orthotropic, and laminated plates for elastic small deflection response are established by comparison with different exact and approximate solutions. The present Dynamic Relaxation (DR) method shows a good agreement with other analytical and numerical methods used in the verification scheme. It was found that: The convergence and accuracy of the DR solution is dependent on several factors which include boundary conditions, mesh size and type, fictitious densities, damping coefficients, time increment and applied load. Also, the DR small deflection program using uniform meshes can be employed in the analysis of different thicknesses for isotropic, orthotropic or laminated plates under uniform loads in a fairly good accuracy.

Comparative performance evaluation of data-driven causality measures applied to brain networks

In this article, several well-known data-driven causality methods are revisited and comparatively evaluated. These are the Granger–Geweke Causality (GGC), the Partial Directed Coherence (PDC), the Directed Transfer Function (DTF) and the Direct Directed Transfer Function (dDTF). The robustness of the four causality measures against two degradation factors is quantitatively evaluated. These are: the presence of realistic biological/electronic noise at various SNR levels, as recorded on a MagnetoEncephalography (MEG) machine, and the presence of a weak node in the brain network where the causality analysis is applied. The causality measures are evaluated in terms of the relative estimation error and the compromise between true and fictitious causal density in the brain network. Both parametric and non-parametric causality analysis is performed. It is illustrated that the non-parametric method is a promising alternative to the more commonly applied MVAR-model based causality analysis. It is also demonstrated that, in the presence of both tested degradation factors, the DTF method is the most robust in terms of low estimation error, while the PDC in terms of low fictitious causal density. The dDTF provides lower fictitious causal density and higher spectral selectivity as compared to DTF, at high enough SNR. The GGC exhibits the worst compromise of performance. An application of the causality measures to a set of MEG resting-state experimental data is accordingly presented. It is demonstrated that significant contrast between the Eyes-Closed and Eyes-Open rest condition in the alpha frequency band allows to detect significant causality between the occipital cortex and the thalamus.