3D Problems Research Articles

Spatial Damage Identification of 3D Structures based on Wavelet-Sparrow Search Algorithm

Based on the singularity principle of the wavelet transform, the continuous wavelet transform is applied to the strain mode of the damaged structure, and the data is normalised and coupled to each 3D position in space. The modal maxima of the wavelet coefficients of each layer of data are compared to identify the spatial location of the damage to the complex structure, and then the Sparrow Search Algorithm (SSA) is used to quickly and accurately identify the data and obtain the degree of damage to the structure. In this paper, the proposed method is validated by numerical finite element simulations and experiments on cantilever structures. The results of the study show that the method can be used to solve the 3D spatial damage identification problem in a simple and fast way, which can be used as a reference for structural inspection problems in practical engineering.

A new implicit gradient damage model based on energy limiter for brittle fracture: Theory and numerical investigation

We present a general form of the gradient-enhanced damage theory and its numerical implementation using finite element method (FEM) in modeling quasi-static brittle crack growth in one- (1D), two- (2D) and three-dimensional (3D) bodies. Coupled equations of the equilibrium and a new implicit gradient damage formulation are introduced to govern the deformation of the solid and evolution of the damage. The resulting nonlocal damage evolution equation featuring the growth of diffusive crack is integrated with a characteristic length scale to eliminate the common mesh-bias issue in FEM implementation. In contrast to the traditional gradient-enhanced damage approaches, the nonlocal damage field here is defined as the primary variable of the damage evolution equation without interpolation mismatch between the displacement and nonlocal damage fields. For derivation of the material constitutive law and local damage parameter, a novel strain energy density (SED) function based on the energy limiter theory for brittle crack growth problems under small strain regime is introduced. To further improve the performance of the developed model, an initial SED threshold, which is used for determining the critical point when damage starts to initiate in the material, is integrated into the novel energy limiter theory. For preventing nonphysical failure in compression domains, the spectral decomposition technique for the strain tensor is adopted to split the reference SED. With integrating the energy limiter into the developed theory, unlike the conventional nonlocal damage theories where the interpretation of the length scale is still ambiguous, the developed nonlocal damage model defines the length scale parameter as the problem-dependent factor. The performance and ability of the proposed model are demonstrated via a set of representative numerical examples in 1D, 2D and 3D fracture problems.

JAX-FEM: A differentiable GPU-accelerated 3D finite element solver for automatic inverse design and mechanistic data science

This paper introduces JAX-FEM, an open-source differentiable finite element method (FEM) library. Constructed on top of Google JAX, a rising machine learning library focusing on high-performance numerical computing, JAX-FEM is implemented with pure Python while scalable to efficiently solve problems with moderate to large sizes. For example, in a 3D tensile loading problem with 7.7 million degrees of freedom, JAX-FEM with GPU achieves around 10× acceleration compared to a commercial FEM code depending on platform. Beyond efficiently solving forward problems, JAX-FEM employs the automatic differentiation technique so that inverse problems are solved in a fully automatic manner without the need to manually derive sensitivities. Examples of 3D topology optimization of nonlinear materials are shown to achieve optimal compliance. Finally, JAX-FEM is an integrated platform for machine learning-aided computational mechanics. We show an example of data-driven multi-scale computations of a composite material where JAX-FEM provides an all-in-one solution from microscopic data generation and model training to macroscopic FE computations. The source code of the library and these examples are shared with the community to facilitate computational mechanics research. Program summaryProgram Title: JAX-FEMCPC Library link to program files:https://doi.org/10.17632/hgwshjbcw6.1Developer's repository link:https://github.com/tianjuxue/jax-am/tree/main/jax_am/femLicensing provisions: GPLv3Programming language: PythonNature of problem: Implementation of the finite element method (FEM) with several appealing features that classic FEM software typically does not have: realized with pure Python; running on CPU/GPU; differentiable for solving PDE-constrained optimization problems; seamless integration with machine learning.Solution method: Our framework JAX-FEM is based on Google JAX, a high-performance numerical computing library with automatic differentiation features and supporting both CPU/GPU. Unlike classic FEM software written in Fortran or C/C++, JAX-FEM is implemented with pure Python and can easily be installed as a Python package. We demonstrate our software by solving problems including forward PDE prediction, inverse design/optimization, and data-driven analysis.

Solving Nonlinear Elliptic PDEs in 2D and 3D Using Polyharmonic Splines and Low-Degree of Polynomials

In this paper, the improved localized method of approximated particular solutions (ILMAPS) using polyharmonic splines (PHS) together with a low-degree of polynomial basis is used to approximate solutions of various nonlinear elliptic Partial Differential Equations (PDEs). The method is completely meshfree, and it uses a radial basis function (RBF) that has no shape parameters. The discretization process is done through a simple collocation technique on a set of points in the local domain of influence. Resulted system of nonlinear algebraic equations is solved by the Picard method.The performance of the proposed method is tested on various nonlinear elliptical problems, including the Poisson-type problems in 2D and 3D with constant or variable coefficients on rectangular or irregular domains and the Poisson–Boltzmann equation with Dirichlet boundary conditions or mixed boundary conditions. The effect of domain shapes in 2D and 3D, types of boundary conditions, and degrees of PHS, and order of polynomial basis are examined. The performance of the method is compared with other bases such as multiquadrics (MQ) basis functions, and with results reported in the literature (method of particular solutions using polynomials). The numerical experiments suggest that ILMAPS with polyharmonic splines yields considerably superior accuracy than other RBFs as well as other approaches reported in the literature for solving nonlinear elliptic PDEs.

Machine-learning assisted topology optimization for architectural design with artistic flavor

A machine-learning assisted topology optimization approach is proposed for architectural design with artistic flavor. This work establishes a novel framework to systematically integrate structural topology optimization with subjective human design preferences. To embed artistic flavor into the design, neural style transfer technique is adopted for measuring and generating the prior knowledge from a reference image with concerned artistic flavor. With the use of different convolutional layers in the VGG-19 (Visual Geometry Group) model-based CNN (Convolutional Neural Network), both style and content of the artistic flavor from low to high levels of abstraction can be constructed. Then, the measured knowledge can be integrated into pixel-based topology optimization as a formal similarity constraint. Both 2D and 3D problems are solved to illustrate the effectiveness of the proposed approach where inheritance of artistic heritage can be achieved in a systematic manner.

Sign stable arbitrary high order reconstructions for constructing non-oscillatory entropy stable schemes

High order sign stable reconstructions are much sought after to develop high order non-oscillatory entropy stable schemes. This work presents a very simple but generic approach to construct arbitrary high order sign stable reconstructions. Further, by utilizing such obtained sign stable reconstructions entropy stable schemes are constructed using well known arbitrary high-order entropy conservative fluxes. Several numerical results for 1D and 2D benchmark test problems are given to demonstrate the accuracy and non-oscillatory property of representative entropy stable schemes which are designed using of proposed approach.

Slope-Scale Rockfall Susceptibility Modeling as a 3D Computer Vision Problem

Rockfall constitutes a major threat to the safety and sustainability of transport corridors bordered by rocky cliffs. This research introduces a new approach to rockfall susceptibility modeling for the identification of potential rockfall source zones. This is achieved by developing a data-driven model to assess the local slope morphological attributes with respect to the rock slope evolution processes. The ability to address “where” a rockfall is more likely to occur via the analysis of historical event inventories with respect to terrain attributes and to define the probability of a given area producing a rockfall is a critical advance toward effective transport corridor management. The availability of high-quality digital volumetric change detection products permits new developments in rockfall assessment and prediction. We explore the potential of simulating the conceptualization of slope-scale rockfall susceptibility modeling using computer power and artificial intelligence (AI). We employ advanced 3D computer vision algorithms for analyzing point clouds to interpret high-resolution digital observations capturing the rock slope evolution via long-term, LiDAR-based 3D differencing. The approach has been developed and tested on data from three rock slopes: two in Canada and one in the UK. The results indicate clear potential for AI advances to develop local susceptibility indicators from local geometry and learning from recent rockfall activity. The resultant models produce slope-wide rockfall susceptibility maps in high resolution, producing up to 75% agreement with validated occurrences.

3D Structural Topology Optimization Using ESO, SESO and SERA: Comparison and an Extension to Flexible Mechanisms

This article investigates the study of Topology Optimization (TO) in 3D elasticity problems to determine the optimal topology by applying the evolutionary methods of Smoothing Evolutionary Structural Optimization (SESO), Sequential Element Rejection and Admission (SERA), and Evolutionary Structural Optimization (ESO). These procedures were implemented in MATLAB code as an extension of Top3d implemented for SIMP by using the eight-node hexahedral finite element formulation in three-dimensional elastostatic structures. The approaches conducted in the present study are demonstrated with numerical examples involving the compliance minimization criterion. Further, a brief synthesis of flexible mechanisms was studied to emphasize the performance of complaint mechanisms measured in terms of two design specifications/functionalities: mechanical and geometrical advantages, which are the highlights of this article. To show the gains of the proposed methods, numerical results obtained are compared with Solid Isotropic Material with Penalization (SIMP) models.

Adaptive hp refinement for spectral elements in numerical relativity

When a numerical simulation has to handle a physics problem with a wide range of time-dependent length scales, dynamically adaptive discretizations can be the method of choice. We present a major upgrade to the numerical relativity code bamps in the form of fully adaptive, physics-agnostic hp refinement. We describe the foundations of mesh refinement in the context of spectral element methods, the precise algorithm used to perform refinement in bamps, as well as several indicator functions used to drive it. Finally, we test the performance, scaling, and the accuracy of the code in treating several 1D and 2D example problems, showing clear improvements over static mesh configurations. In particular, we consider a simple nonlinear wave equation, the evolution of a real scalar field minimally coupled to gravity, as well as nonlinear gravitational waves.

A scaled boundary finite element method for soil dynamic impedance of pile groups using hybrid quadtree mesh considering horizontal vibration

The numerical modeling of pile-soil dynamic interaction has been a challenging problem in engineering for many years because of the unbounded domain, which typically involves significant computational cost. This paper develops a novel scaled boundary finite element method (SBFEM) that can accurately calculate the soil dynamic impedance of pile groups with end bearing subjected to horizontal vibration. The three-dimensional (3D) problem is first transformed to a two-dimensional (2D) Helmholtz equation in the horizontal plane, exploiting the analytical solution for the vertical modes using separation of variables. The SBFEM is then applied to solve the 2D computational domain, which is divided into bounded and unbounded domains. Based on the SBFEM, an accurate circular artificial boundary condition (ABC) is formulated to simulate the unbounded domain of the soil. A special hybrid quadtree mesh is devised for this problem, where a high-quality structured quadrilateral mesh is generated surrounding the piles. The accuracy of the proposed method is verified using a single pile problem with known analytical solution. The effects of dimensionless frequency, pile numbers, soil depth and relative distance between piles on the soil dynamic impedance are considered.

Numerical study of heat transfer and melting process in a NEPCM filled 3D cuboid cavity: Application non-uniform temperature

In this numerical study, a cubic box filled with a Nanoparticle Enhanced Phase Change Material (NEPCM) (mixture “N-Octadecane/alumina (Al2O3)"), is considered. Using Comsol-Multiphysics program, the configuration is treated as a 3D symmetric numerical simulation problem under fixed boundary conditions. Besides, heat transfer fluid (HTF) generates a non-uniform temperature that varies linearly along an internal tube installed in the lowest area of the enclosure to ensure the melting of the NEPCM. Therefore, the purpose of the study is to highlight the effect of the nanoparticles volume fraction (varying from 0 to 6%) on thermal distribution and melting performances. The obtained findings figure out that the PCM melts irregularly and is affected by the non-uniform temperature of the HTF tube. Hence, the third dimension should be considered, particularly in PCM ring tank simulations, in order to obtain reliable predictions compared to practical and experimental studies. Additionally, the Al2O3 nanoparticles have a valuable effect on improving the thermal conductivity of NEPCM, where thermal resistance and melting time are reduced with the increase of the nanoparticles volume fraction. Compared to the case of pure PCM, the heat transfer enhancement yields to an augmentation in the melting rate by about 11% and 9.5% for the volume fractions of 6% and 3%, respectively. Moreover, there is a significant increase in the amount of stored thermal energy when nanoparticles are dispersed in the PCM.

A non-iterative methodology to reconstruct boundary shapes and conditions in isotropic linear elasticity based on the BEM

A non-iterative method is proposed to estimate unknown boundary geometry shapes and boundary conditions in 2D and 3D isotropic linear elasticity problems based on the boundary element method (BEM). A virtual domain with a virtual boundary is introduced to the inverse problem. For the boundary geometry shape identification problem, the displacements of the virtual boundary are obtained through the measurement point displacements and the singular value decomposition is applied to solve the ill-posed matrix. The BEM is used to solve the problem with the displacements of the virtual boundary. The unknown boundary geometry shapes can be estimated by searching the internal point isodisplacement curve. For the boundary condition reconstruction problem, the virtual boundary is overlapped with the real boundary. The displacements of the virtual boundary can also be acquired by the measurement point displacements. The unknown boundary tractions can be calculated by the BEM. Finally, the influences with different virtual domains, measurement points, measurement positions and measurement errors on the results are discussed. The results show that the proposed method is an accurate and efficient way to identify boundary geometry shapes and reconstruct boundary conditions.

Intelligent flying-beamformer for hybrid mmWave systems: A deep reinforcement learning approach

Millimeter wave (mmWave) enabled unmanned aerial vehicle (UAV) communications with excellent flexibility and high data transmission capabilities are widely regarded as an essential element of the next generation non-terrestrial networks. In this paper, we investigate the 3D trajectory design, beamwidth and power allocation problems for mmWave UAV communication systems. We first formulate a non-convex optimization problem to maximize the total normalized spectral efficiency (NSE) of all ground terminals (GTs). Then, we propose a deep reinforcement learning (DRL) based framework, termed as intelligent flying-beamformer, to solve the formulated non-convex optimization, which contains of two sequential phases: 3D UAV trajectory design, and joint optimization of beamwidth and power allocation. In particular, the off-policy deep deterministic policy gradient (DDPG) and the on-policy proximal policy optimization (PPO) algorithms are used to process the complex data in the two phases, respectively. Simulation results verify the effectiveness of the proposed intelligent flying-beamformer in improving system capacity. Specifically, numerical results show that the optimization of the beamwidth can indeed significantly improve the spectral efficiency of mmWave UAV network.

Families of Orbits Produced by Three-Dimensional Central and Polynomial Potentials: An Application to the 3D Harmonic Oscillator

We study three-dimensional potentials of the form V=U(xp+yp+zp), where U is an arbitrary function of C2-class, and p∈Z, which produces a preassigned two-parametric family of spatial regular orbits given in the solved form f(x,y,z) = c1, g(x,y,z) = c2 (c1, c2 = const). These potentials have to satisfy two linear PDEs, which are the basic equations of the 3D inverse problem of Newtonian dynamics. The functions f and g can be represented uniquely by the ”slope functions” α(x,y,z) and β(x,y,z). The orbital functions α(x,y,z) and β(x,y,z) have to satisfy three differential conditions according to the theory of the inverse problem. If these conditions are satisfied, then we can find such a potential analytically. We offer pertinent examples of potentials that are mainly used in physical problems. The values obtained for p lead to cases of well-known potentials, such as the Newtonian, cored, logarithmic, polynomial and quadratic ones. New families of orbits produced by the 3D harmonic oscillator are found. Pertinent examples are given and cover all cases. Two-dimensional potentials belong to a special category of potentials and are studied separately. The families of straight lines in 3D space are also examined.

A numerical two-scale approach for nonlinear hyperelastic beams and beam networks

We introduce a numerical framework for modeling hyperelastic slender trusses, oftentimes used as elementary building blocks in architected materials, which accounts for both the geometric nonlinearity inherent in thin structures and the nonlinear constitutive behavior of the base material. Akin to the FE2 method in homogenization, our approach is based on a formal, two-scale expansion. We decompose the three-dimensional (3D) description of a slender structure into a macroscale problem solving for the deformation of the beam center-line (based on an effective strain energy density, which depends on the stretching, bending, and torsional strains of the beam’s center-line) and a series of two-dimensional (2D) microscale boundary value problems (defined over the beam cross-section). We solve a series of 2D problems (covering a range of macroscopic strain combinations) for a given cross-sectional geometry and material distribution in an offline, pre-processing step. Using this pre-computed energy landscape, we solve the macroscopic boundary value problem within a geometrically exact, nonlinear discrete beam framework. We demonstrate the accuracy of this approach through a set of benchmark problems highlighting the nonlinear effects of the cross-sectional geometry and constitutive material, including material heterogeneity and pre-strains. We further illustrate how this technique is applied to truss-based architected materials consisting of networks of slender struts, which are typically made of polymeric base materials and thus allow for large nonlinear elastic deformation.

Response of elastic/viscoelastic layers on an elastic half-space in rolling contacts: Towards a new modeling approach for elastohydrodynamic lubrication

The 3D quasi-static contact problem of a rigid sphere rolling on an elastic half-space covered by a thin viscoelastic/elastic layer is studied as representative of soft layered contacts in engineering, physics, and biomedical applications, as well as for its potential merit to model Elasto-Hydrodynamically Lubricated (EHL) contacts in specific EHL film behavior suggested in earlier research. The viscoelastic layer behavior is modeled with a standard linear solid (SLS) model with a single relaxation time. Two approaches are used, the foundation (Winkler) reduced model assuming unidirectional stress–strain behavior normal to the surface only and the Papkovich–Neuber potential model which, based on the complete Navier–Cauchy equations, accounts for non-local support and bending effects. The models are validated against literature and compared. Whereas coated/layered contact problem studies mostly consider relatively thick coatings, and results focus on the pressure distribution and the contact area size, in this paper we consider layers with a thickness much smaller than the contact radius, i.e. of the order of a thickness of a conventional EHL film. The details of the layer deformation, pressure profiles and subsurface stresses are presented and interpreted in terms of the underlying physics, in particular the thin layer limit common also in thin fluid layers. It is shown that two dimensionless parameters, the ratio of the elastic modulus between the layer and the substrate and the ratio of the layer thickness to the corresponding Hertz contact width, dominate the systematic response for elastic layered contact problems. And four dimensionless parameters with two extras, the Deborah number based on the Hertz contact width and the ratio of the two elastic limits of the SLS viscoelastic material, are needed and analyzed for viscoelastic layered contacts. The results presented provide a good framework for the understanding and interpretation of the effects of viscoelastic layers on the deformation and pressure distribution of contact problems. Finally, the capability of actually modeling EHL film behavior via a viscoelastic layer suggested by van Emden et al. (2017) is discussed.

Coupling physics-informed neural networks and constitutive relation error concept to solve a parameter identification problem

Identification of material model parameters using full-field measurement is a common process both in industry and research. The constitutive equation gap method (CEGM) is a very powerful strategy for developing dedicated inverse methods, but suffers from the difficulty of building the admissible stress field. In this work, we present a new technique based on physics-informed neural networks (PINNs) to implement a CEGM optimization process. The main interest is to easily construct the admissible stress thanks to automatic differentiation (AD) associated with PINNs. This new method combines the high quality of the CEGM with the numerical effectivity of the PINNs and realizes the identification of material properties in a more concise way. We compare two variants of the developed method with the classical identification strategies on simple two-dimensional (2D) cases and illustrate its effectiveness in three-dimensional (3D) problems, which is of interest when dealing with tomographic images. The results indicate that the proposed method has good performance while avoiding complex calculation procedures, showing its great potential for practical applications.

A model order reduction technique for the linearised compressible flow equations

AbstractViscous effects in acoustics are significant if the thickness of the oscillating viscous boundary layer (Stokes boundary layer) becomes comparable to characteristic problem dimensions. For air as an acoustic medium, this is frequently the case in small‐scale devices like micro‐electro‐mechanical systems or micro‐perforated panels for sound absorption. Accurate modelling of viscous effects can be done by the solution of the linearised compressible flow equations by the finite element method, which is computationally demanding due to the high number of unknowns arising in the problem: pressure, velocity and temperature. Furthermore, the discretisation must resolve the thin viscous boundary layers, creating a model with a high number of degrees of freedom. We suggest a projection‐based model order reduction procedure using the system eigenmodes as generalised coordinates. The procedure is tested based on a simple example problem in 2D. Here we show that the derived reduced order model is accurate and computationally highly efficient. The number of degrees of freedom can be reduced by several orders of magnitude, from around 30 000 to 100, thereby dramatically reducing the computation time for harmonic solutions.

A particle finite element method based on Level–Set functions

Since the seminal work of Idelsohn, Oñate and Del-Pin (2004), the Particle Finite Element Method (PFEM) has relied on a Delaunay triangulation and the Alpha–Shape (AS) algorithm in the remeshing process. This approach guarantees a good quality of the Lagrangian mesh, but introduces a list of shortcomings that demand geometrical treatments tailored to each problem. In order to improve the remeshing process in PFEM, this work proposes the use of a Level–Set (LS) function instead of the Alpha–Shape algorithm. Since the Level–Set considers the boundary of the fluid and its interior, and not only a geometric criterion as does the Alpha–Shape, the proposed strategy (PFEM–LS) shows more robustness than the classical approach (PFEM–AS) owing to three main improvements. First, the LS function allows for a better control over the elements that are created during the fluid/fluid contact, which helps to reduce mass creation. Second, it helps to preserve the smoothness of the free surface and to reduce mass loss. Third, it allows the meshing of solitary particles that are detached from the free surface, which improves the representation of drops in PFEM. The methodology is presented and validated using free surface flow problems in 2D.

An Evaluation of Accuracy and Efficiency of a 3D Adaptive Mesh Refinement Method with Analytical Velocity Fields

Appropriate mesh refinement plays a vital role in the accuracy and convergence of computational fluid dynamics solvers. This work is an extension of the previous work that further demonstrates the accuracy of the 3D adaptive mesh refinement method by comparing the accuracy measures between the ones derived from the analytical fields and those identified by the refined meshes. The adaptive mesh refinement method presented in this study is based on the law of mass conservation for three-dimensional incompressible or compressible steady fluid flows. The assessment of the performance of the adaptive mesh refinement method considers its key features such as drawing closed streamline and identification of singular points, asymptotic planes, and vortex axis. Several illustrative examples of the applications of the 3D mesh refinement method with a multi-level refinement confirm the accuracy and efficiency of the proposed method. Furthermore, the results demonstrate that the adaptive mesh refinement method can provide accurate and reliable qualitative measures of 3D computational fluid dynamics problems.