3D Problems Research Articles

Solid Phase Transitions in the Liquid Limit

We address the fundamental difference between solid-solid and liquid-liquid phase transitions within the Ericksen's nonlinear elasticity paradigm. To highlight ideas, we consider the simplest nontrivial 2D problem and work with a prototypical two-phase Hadamard material which allows one to weaken the rigidity and explore the nature of solid-solid phase transitions in a ``near-liquid'' limit. In the language of calculus of variations we probe limits of quasiconvexity in an ``almost liquid'' solid by comparing the thresholds for cooperative (laminate based) and non-cooperative (inclusion based) nucleation. Using these two types of nucleation tests we obtain for our model material surprisingly tight two-sided bounds on the elastic binodal without directly computing the quasiconvex envelope.

Three-Dimensional Tracking of a Target under Angle-Frequency Measurements with Multiple Frequency Lines.

This article considers tracking a constant-velocity underwater target, which emits sound with distinct frequency lines. By analyzing the target's azimuth, elevation and multiple frequency lines, the ownship can estimate the target's position and (constant) velocity. In our paper, this tracking problem is called the 3D Angle-Frequency Target Motion Analysis (AFTMA) problem. We consider the case where some frequency lines disappear and appear occasionally. Instead of tracking every frequency line, this paper proposes to estimate the average emitting frequency by setting the average frequency as the state vector in the filter. As the frequency measurements are averaged, the measurement noise decreases. In the case where we use the average frequency line as our filter state, both the computational load and the root mean square error (RMSE) decrease, compared to the case where we track every frequency line one by one. As far as we know, our manuscript is unique in addressing 3D AFTMA problems, such that an ownship can track an underwater target while measuring the target's sound with multiple frequency lines. The performance of the proposed 3D AFTMA filter is demonstrated utilizing MATLAB simulations.

Stress intensity factor and fatigue life evaluation for important points of semi-elliptical cracks in welded pipeline by Bezier extraction based XIGA and new correlation model

NURBS-based Bezier extraction in extended isogeometric analysis (XIGA) using thermo elastic-plastic equation has been proposed for evaluating the stress intensity factor (SIF) in the various relative crack depths, aspect ratios, and important points (surface points and deepest point) of the semi-elliptical surface crack in weld line of pipelines under cyclic internal pressure. In this approach, the level set technique is used for identifying the grid points around the crack front and the crack surface. Crack in the specified domain is modeled by Heaviside and crack front enrichment functions and remeshing due to crack growth does not require. Also, new correlation models are proposed to predict SIF in these points of the crack front. Based on the proposed correlations, fatigue crack propagation, and fatigue life are investigated. Modified walker equation used for predicting fatigue crack propagation rate and fatigue life in the welded pipeline. The finite element method (FEM) analysis with a similar condition was performed and compared with XIGA and correlation models to validate the presented method. The results demonstrate good agreement. Thus, the proposed approach successfully studies the 3D crack problem with acceptable accuracy.

Thermal–mechanical metamaterial analysis and optimization using an Abaqus plugin

This paper demonstrates how commercial finite-element software and optimization algorithms can be combined to fully explore the design space of thermal–mechanical metamaterials to reveal trends and new insight. This is achieved by developing an Abaqus plugin (EasyPBC) that automates the application of periodic boundary conditions and computes effective elastic and thermal expansion properties for 2D and 3D problems. Abaqus is then linked to an optimizer to fully explore the design space and optimal trade-off between thermal and mechanical properties for two example metamaterials. The first example is a auxetic 2D star-shaped metamaterial, where the proposed approach is used to create a design envelope for Poisson’s ratio and thermal expansion coefficient by solving a series of constrained optimization problems. The second example is a 3D metamaterial based on an octet truss, with additional members to expand the design space. A multi-objective optimization problem is solved to find the optimal trade-off between Young’s modulus and thermal expansion coefficient in a prescribed direction. The results of both examples expand our knowledge about the range of properties for these metamaterials, and designs for optimal trade-off between thermal and mechanical properties.

Supercloseness of weak Galerkin method for a singularly perturbed convection–diffusion problem in 2D

In this paper we analyze a weak Galerkin method on a Shishkin rectangular mesh for a singularly perturbed convection–diffusion problem in two dimensions, whose solution has one exponential layer and two parabolic layers. The supercloseness of this method is achieved by a specially constructed interpolant. More specifically, the interpolant defined in the interior of each element consists of a vertices-edges-element interpolant inside the layers and a modified Gauß-Radau interpolant outside the layers, while the interpolant defined on the boundary of each element consists of a vertices-edges-element interpolant inside the layers and a weighted L2 projection outside the layers. Moreover, we make use of the over-penalization technique inside the layers, and then prove supercloseness of order k+1/2, even up to almost k+1 under appropriate assumptions. Numerical experiments support the theoretical result.

Surface penalization of self-interpenetration in linear and nonlinear elasticity

We analyze a term penalizing surface self-penetration, as a soft constraint for models of hyperelastic materials to approximate the Ciarlet-Nečas condition (almost everywhere global invertibility of deformations). For a linear elastic energy subject to an additional local invertibility constraint, we prove that the penalized elastic functionals converge to the original functional subject to the Ciarlet-Nečas condition. The approach also works for nonlinear models of non-simple materials including a suitable higher order term in the elastic energy, without artificial local constraints. Numerical experiments illustrate our results for a self-contact problem in 3d.

A PDE-constrained optimization method for 3D-1D coupled problems with discontinuous solutions

AbstractA numerical method for coupled 3D-1D problems with discontinuous solutions at the interfaces is derived and discussed. This extends a previous work on the subject where only continuous solutions were considered. Thanks to properly defined function spaces a well posed 3D-1D problem is obtained from the original fully 3D problem and the solution is then found by a PDE-constrained optimization reformulation. This is a domain decomposition strategy in which unknown interface variables are introduced and a suitably defined cost functional, expressing the error in fulfilling interface conditions, is minimized constrained by the constitutive equations on the subdomains. The resulting discrete problem is robust with respect to geometrical complexity thanks to the use of independent discretizations on the various subdomains. Meshes of different sizes can be used without affecting the conditioning of the discrete linear system, and this is a peculiar aspect of the considered formulation. An efficient solving strategy is further proposed, based on the use of a gradient based solver and yielding a method ready for parallel implementation. A numerical experiment on a problem with known analytical solution shows the accuracy of the method, and two examples on more complex configurations are proposed to address the applicability of the approach to practical problems.

EMFEM: A parallel 3D modeling code for frequency-domain electromagnetic method using goal-oriented adaptive finite element method

We have developed an open-source parallel 3D forward modeling code called EMFEM for the frequency-domain electromagnetic method (FDEM). This code is based on the adaptive finite element method and uses the total-field approach to solve the time-harmonic Maxwell equations. The use of tetrahedral meshes enables modeling of problems with complex geometry and topography. We utilize an iterative solver and an optimal block-diagonal preconditioner and auxiliary Maxwell solver to accelerate the convergence rate. We also use goal-oriented error estimation to adaptively refine the mesh and improve the accuracy of the solution. In addition, we develop a parallel algorithm based on the mesh partitioning approach. The effectiveness of the adaptive mesh refinement is demonstrated through a 1D layered model for marine controlled-source electromagnetic (CSEM) problems. We further present two 3D models for CSEM and magnetotelluric (MT) problems to study the scalability of the parallel algorithm, and the accuracy of the solutions is cross-validated with other open-source codes. The code is implemented in C++ and follows the modular design principle, making it easy to extend and maintain. The code is capable of modeling large-scale 3D problems and utilizing high-performance computing platforms. Furthermore, as the code is freely available, it can serve as a cross-validation tool for other 3D modeling codes and can also be utilized to implement 3D inversion routines.

An improved elastic wavefield separation method based on the Helmholtz decomposition

Elastic wave-mode separation plays an important role in elastic reverse time migration and elastic full-waveform inversion. It helps to remove crosstalk artifacts and improve imaging quality. A class of efficient methods for elastic wave-mode separation in isotropic elastic media is the Helmholtz decomposition technique. Although this kind of approach produces pure-mode vector wavefields with correct amplitudes, phases, and physical units, their computational costs are still high especially for 3D large-scale problems. By making use of the relationships among the divergence, curl, gradient, and exterior derivative operations, we develop an improved elastic wave-mode separation based on the Helmholtz decomposition. We also need to solve a Poisson equation, but the Laplace operator operates on a scalar function rather than a vector function. Thus, for multidimensional (2D or 3D) problems, the Poisson equation only needs to be solved once for the vector P and S wavefields. This allows us to reduce the computational cost of the conventional Helmholtz decomposition method by a factor of two for solving 2D problems. For a 3D problem, the computational cost can be reduced by a factor of three. To further reduce the computational cost, by introducing a smooth extension technique, we transform the problem into the wavenumber domain via the Fourier transform and use a fast solver for the Poisson equation. The resulting wavefield separation method not only produces P and S waves with the same phases and amplitudes as the input-coupled wavefields but also significantly reduces the computational cost. Numerical tests indicate the efficiency of the proposed method and confirm the theoretical results.

A meshfree method for the solution of 2D and 3D second order elliptic boundary value problems in heterogeneous media

We propose a simple meshfree method for two-dimensional and three-dimensional second order elliptic boundary value problems in heterogeneous media based on equilibrated basis functions. The domain is discretized by a regular nodal grid, over which the degrees of freedom are defined. The boundary is also discretized by simply introducing some boundary points over it, independent from the domain nodes, granting the method the ability of application for arbitrarily shaped domains without the drawback of irregularity in the nodal grid. In heterogeneous media, the governing Partial Differential Equation (PDE) has non-constant coefficients for the partial derivatives, preventing Trefftz-based techniques to be applicable. The proposed method satisfies the PDE independent from the boundary conditions, as in Trefftz approaches. Meanwhile, by applying the weak form of the PDE instead of the strong form through weighted residual integration, the inconvenience of variable coefficients shall be resolved, since the bases will not need to analytically satisfy the PDE. The weighting in the weak form integrals is such that the boundary integrals vanish. The boundary conditions are thus collocated over the defined boundary points. Domain integrals break into algebraic combination of one-dimensional pre-evaluated integrals, thus omitting the numerical integration from the solution process. Each node corresponds to a local sub-domain called cloud. The overlap between adjacent clouds ensures integrity of the solution and its derivatives throughout the domain, an advantage with respect to C0 formulations.

Machine learning‐based prediction of the seismic response of fault‐crossing natural gas pipelines

AbstractHerein, we utilized machine‐learning (ML) and data‐driven (regression) techniques to tackle a critical infrastructure engineering problem—namely, predicting the seismic response of natural gas pipelines crossing earthquake faults. Such a 3D nonlinear problem can take up to 10 h to solve by performing finite element analysis (FEA), considering the length of the pipeline and a large number of pipe and soil elements. However, the ML and data‐driven techniques can learn the projection rule of input‐output and predict the pipeline response instantaneously given a set of input features. In addition, the well‐trained ML model can be implemented for regional‐scale risk and rapid post‐event damage assessments. In this study, the input for ML comprised approximately 217K nonlinear FEAs, which covered a wide range of combinations of soil, structural and fault properties and yielded critical pipe strain responses under fault‐rupture displacements. We adopted various regression models and physics‐constrained neural networks, which can accurately and rapidly predict the tensile and compressive strains for a broad range of probable fault‐rupture displacements. Performances of various ML and conventional statistical models were systematically examined. Not surprisingly, neural networks exhibited the best performance for this multi‐output regression problem, in whichR2 > 0.95 was achieved for a wide range of fault displacement (FD) levels. Further, we used the trained neural network with 14.5 million Monte‐Carlo‐generated input samples to predict the maximum tensile and compressive strain curves of pipelines. This new dataset aimed at filling the missing input‐output points from the 217K FEAs, and improved the accuracy of the prediction of probability of failure for natural gas pipelines under FD hazards.

An Introduction to Programming Physics-Informed Neural Network-Based Computational Solid Mechanics

Physics-informed neural network (PINN) has recently gained increasing interest in computational mechanics. This work extends the PINN to computational solid mechanics problems. Our focus will be on the investigation of various formulation and programming techniques, when governing equations of solid mechanics are implemented. Two prevailingly used physics-informed loss functions for PINN-based computational solid mechanics are implemented and examined. Numerical examples ranging from 1D to 3D solid problems are presented to show the performance of PINN-based computational solid mechanics. The programs are built via Python with TensorFlow library with step-by-step explanations and can be extended for more challenging applications. This work aims to help the researchers who are interested in the PINN-based solid mechanics solver to have a clear insight into this emerging area. The programs for all the numerical examples presented in this work are available at https://github.com/JinshuaiBai/PINN_Comp_Mech .

DEM study of particle scale effect on plain and rotary jacked pile behaviour in granular materials

The capacity of open-ended piles strongly depends on the potential plugging occurring during installation. The Discrete Element Method (DEM) is particularly suitable to the study of pile plugging, as it can model large soil deformation. However, DEM simulation of the 3D boundary value problems is computationally expensive, and particle upscaling is usually used to reduce the number of particles modelled. In this paper, two granular beds were created with the same average porosity, initial stress field and contact parameters, but different particle scales. Two pile geometries were installed by plain and rotary jacking. Normalised results show that the pile penetration mechanism is strongly affected by the particle scale. Larger particles lead to earlier pile plugging, higher shaft resistance and require a greater force to penetrate the ground. This effect can be linked to a modified penetration mechanism, with larger shear zones and less well defined “nose cone” under the pile tip for larger particles. Changing particle scaling has a neutral effect on the total penetration resistance of a rotary jacked pile, but reduces the base penetration and increases the plug resistance. Maximising the ratio of particle scale to wall thickness is key to adequately capturing the pile coring mechanism.

An efficient heterogeneous parallel algorithm of the 3D MOC for multizone heterogeneous systems

Recent innovations in powerful high-performance computing (HPC) architecture have enabled high-fidelity whole-core neutron transport simulations in a reasonable amount of time. Among high-fidelity numerical methods, the one-step 3D Method of Characteristic (MOC) is becoming increasingly popular in the reactor design community due to its numerical accuracy, geometrical flexibility and high degrees of parallelism. In this work, we propose a sophisticated heterogeneous parallel algorithm of the 3D MOC based on a supercomputer prototype consisting of brand-new multizone heterogeneous processors called the MT-3000, which has a peak double precision performance of 11.6 TFLOPS when operating at 1.2 GHz. In the 3D MOC calculation procedure, the total run time is dominated by the so-called MOC transport sweep where large amounts of tracks traverse the mesh grids along different directions. We significantly improve the performance of the MOC transport sweep through a three-level parallel algorithm with multiple hiding latency techniques. We have also implemented heterogeneous level-1 basic linear algebra subprograms (BLAS) that's the second computational hot spot. The 3D C5G7 benchmark problem is employed to verify the correctness and efficiency of the heterogeneous parallel 3D MOC algorithm. Numerical results show that it achieves very high performance on the MT-3000-based supercomputer prototype. Specifically, the three-layer parallel algorithm has proven to be highly scalable for each layer, and the Quarter-MT-3000 performance of the MOC transport sweep has achieved 23.6% of the peak performance, which is close to the theoretical upper limit.

Violation of Neumann Problem’s Solvability Condition for Poisson Equation, Involved in the Non-Linear PDEs, Containing the Reaction-Diffusion-Convection-Type Equation, at Numerical Solution by Direct Method

In this paper, we consider the 3D problem of laser-induced semiconductor plasma generation under the action of the optical pulse, which is governed by the set of coupled time-dependent non-linear PDEs involving the Poisson equation with Neumann boundary conditions. The main feature of this problem is the non-linear feedback between the Poisson equation with respect to induced electric field potential and the reaction-diffusion-convection-type equation with respect to free electron concentration and accounting for electron mobility (convection’s term). Herein, we focus on the choice of the numerical method for the Poisson equation solution with inhomogeneous Neumann boundary conditions. Despite the ubiquitous application of such a direct method as the Fast Fourier Transform for solving an elliptic problem in simple spatial domains, we demonstrate that applying a direct method for solving the problem under consideration results in a solution distortion. The reason for the Neumann problem’s solvability condition violation is the computational error’s accumulation. In contrast, applying an iterative method allows us to provide finite-difference scheme conservativeness, asymptotic stability, and high computation accuracy. For the iteration technique, we apply both an implicit alternating direction method and a new three-stage iteration process. The presented computer simulation results confirm the advantages of using iterative methods.

New aspects of the CISAMR algorithm for meshing domain geometries with sharp edges and corners

Several new algorithmic aspects of the conforming to interface structured adaptive mesh refinement (CISMAR) mesh generation technique are presented for creating high-fidelity finite element models of 3D problems with non-smooth (C0–continuous) geometries. The Stereolithography (STL) file characterizing the domain geometry is first pre-processed to identify its sharp edges and corners, followed by overlaying that on a structured background mesh composed of tetrahedral elements to identify elements cuts by material interfaces/boundaries. Three non-iterative algorithms, namely hierarchical r-adaptivity, element deletion, and Kirigami-inspired sub-tetrahedralization are then introduced to generate a high-quality conforming mesh by relocating background mesh nodes, eliminating degenerate elements, and subdividing non-conforming tetrahedra to match material interfaces and domain boundaries. These algorithms either upgrade or fully replace different phases of the original CISAMR technique, which was limited to modeling problems with smooth interfaces. The example problems provided in this manuscript demonstrate the upgraded CISAMR algorithm to handle domain geometries with complex, non-smooth interfaces.

Application of Adapted-Bubbles to the Helmholtz Equation with Large Wavenumbers in 2D

An adapted bubble approach which is a modifiation of the residual-free bubbles (RFB) method, is proposed for the Helmhotz problem in 2D. A new two-level finite element method is introduced for the approximations of the bubble functions. Unlike the other equations such as the advection-diffusion equation, RFB method when applied to the Helmholtz equation, does not depend on another stabilized method to obtain approximations to the solutions of the sub-problems. Adapted bubbles (AB) are obtained by a simple modification of the sub-problems. This modification increases the accuracy of the numerical solution impressively. The AB method is able to solve the Helmholtz equation efficiently in 2D up to ch = 3.5 where c is the wave number and h is the mesh size. We provide analysis to show how the AB method mitigates the pollution error.

Streamline-based reactive transport modeling of uranium mining during in-situ leaching: Advantages and drawbacks

Reactive transport modeling is known to be computationally intensive when applied to 3D problems. Transforming sequential computing on the computer processor units (CPU) into parallelized computation on the high-performance parallel graphic processor units (GPU) is a classical approach to increasing computational performance. Another complementary approach is to decompose a complex 3D modeling problem into a set of simpler 1D problems using streamline approaches which can be easily parallelized, therefore reducing computation time. This paper investigates solutions to the equations governing dissolution and transport using streamlines coupled with a parallelization approach. In addition, an analytical solution to the dissolution and transfer equations of uranium describing the In-Situ Leaching (ISL) mining recovery is found using an approximation series to the 2nd order. The analytical solution is compared to the 1D numerical resolution along the streamlines and to the 3D simulation results superimposed on the streamline. Both approaches give similar results with a relative error of <2 % (2%). The proposed methodology is then applied to a case study in which the classical 3D resolution is compared to the newly suggested streamline solution, demonstrating that the streamline approach increases computational performances by a factor ranging from hundred to thousand depending on the complexity of the grid-block model.

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.