2D Examples Research Articles

HiDeNN-FEM: A seamless machine learning approach to nonlinear finite element analysis.

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

Interactive Structural Topology Optimization with Subjective Scoring and Drawing Systems

Topology optimization techniques can create efficient and innovative structural designs by redistributing underutilized materials to the most-needed locations. These techniques are typically performed based purely on structural performance without considering factors like aesthetics and other design requirements. Hence, the obtained structural designs may not be suitable for specific practical applications. This study presents a new topology optimization method, SP-BESO, by considering the subjective preferences (SP) of the designer. Here, subjective scoring and drawing systems are introduced into the popular bi-directional evolutionary structural optimization (BESO) technique. The proposed SP-BESO method allows users to iteratively and interactively create topologically different and structurally efficient solutions by explicitly scoring and drawing their subjective preferences. Hence, users do not need to passively accept the optimization results. A user-friendly digital design tool, iBESO, is developed, which contains four optimizers to simultaneously perform the proposed SP-BESO method to assist in the design exploration task. A variety of 2D examples are tested using the iBESO software to demonstrate the effectiveness of the proposed SP-BESO method. It is found that the combination of parameters used in the scoring and drawing systems controls the formation of final structural topologies toward performance-driven or preference-driven designs. The utilization of the proposed SP-BESO method in potential practical applications is also demonstrated.

An iterative approach using Sawi transform for fractional telegraph equation in diversified dimensions

Abstract In the present study, 1D, 2D, and 3D fractional hyperbolic telegraph equations in Caputo sense have been solved using an iterative method using Sawi transform. These equations serve as a model for signal analysis of electrical impulse transmission and propagation. Along with a table of Sawi transform of some popular functions, some helpful results on Sawi transform are provided. To demonstrate the effectiveness of the suggested method, five examples in 1D, one example in 2D, and one example in 3D are solved using the proposed scheme. Error analysis comparing approximate and exact solutions using graphs and tables has been provided. The proposed scheme is robust, effective, and easy to implement and can be implemented on variety of fractional partial differential equations to obtain precise series approximations.

3D true-amplitude elastic wave-vector decomposition in heterogeneous anisotropic media

Wave-mode separation aims at separating the elastic wavefield into P- and S-wave modes in each subsurface grid. Several wave-mode separation methods require the calculation of polarization vectors of different wave modes. We have developed a wave-mode separation method based on an analytical decomposition operator and a low-rank approximation in the wavenumber domain. When dealing with a general anisotropic medium with low symmetry, the two S-wave modes suffer from the singularity problem, where strong planar artifacts are caused due to the discontinuous polarization vectors at the singularities. A weight function with an adjustable threshold parameter can be designed to mitigate the strong artifacts, thereby obtaining clean S-wave modes. Larger threshold results in a stronger suppression of the artifacts but at the expense of stronger damages to S-wave energy. Here, we provide a new way to deal with the planar artifacts. The S-wave modes separated using a zero threshold containing strong artifacts can be treated as a noisy wavefield. Applying a weighting function can be viewed as a denoising process. Choosing the threshold parameter in the weighting function will inevitably cause an amplitude loss of the signal. Thus, we can apply the local orthogonalization method to compensate for the amplitude loss. Considering the heterogeneity of the wavefield, we apply a nonstationary local orthogonalization method to obtain an accurate wave-mode separation. The final separated wavefields are true amplitude and thus kinematically and dynamically correct, which will benefit a variety of seismic forward and inverse problems. Several 3D synthetic examples demonstrate the performance of the new method.

Solution and sensitivity analysis of nonlinear equations using a hypercomplex-variable Newton-Raphson method

The classical Newton-Raphson (NR) method for solving nonlinear equations is enhanced in two ways through the use of hypercomplex variables and algebra. In particular, i) the Jacobian is computed in a highly accurate and automated way, and ii) the derivative of the solution to the nonlinear equations is computed with respect to any parameter contained within the system of equations. These advances provide two significant enhancements in that it is straightforward to provide an accurate Jacobian and to construct a reduced order model (ROM) of arbitrary order with respect to any parameter of the system. The ROM can then be used to approximate the solution for other parameter values without requiring additional solutions of the nonlinear equations. Several case studies are presented including 1D and 2D academic examples with fully functioning Python code provided. Additionally, a case of study of the catenary of an elastic cable subject to its own weight and a vertical point load. Derivatives up to 10th order were computed with respect to material, loading, and geometrical parameters. The derivatives were used to generate reduced order models of the cable deformation and reaction forces at its ends with respect to multiple input parameters. Results show that from a single hypercomplex evaluation of the cable under a single vertical point load, it is possible to generate an accurate reduced order model capable of predicting the cable deformation with 1.5 times the load in the opposite direction and with 3.5 times the load in the same direction without resolving the system of equations.

Modeling the frequency response of an acoustic cavity using the method of images

We demonstrate the ability of a simple algorithm based on the venerable method of images (MOIs), to accurately model the detailed frequency response of a multidimensional, rectangular, lossy resonant cavity. The convergence properties of the model's infinite series solution are shown to be determined by the cavity's quality factor Q. A 1D example demonstrates that the MOI series converges to the exact solution. Next, a comparison to precisely measure 2D cavity data confirms that a straightforward extension of the 1D algorithm to multiple dimensions provides accurate results. The algorithm is short, easily understandable by undergraduate students and relatively undemanding to code. An example using ®mathematica is provided.

Finding globally optimal arrangements of multiple point loads in structural design using a single FEA

Many structures around us are designed to carry point loads. Such structures are typically sensitive to load arrangements, including load locations, magnitudes, and directions; a slight change in these ingredients could significantly affect the structural response. Therefore, knowing the extremal load arrangements to achieve the best and worst structural performance holds great potential to maximize structural efficiency and avoid structural failure, respectively. Existing studies have attempted to optimize load conditions using iterative optimization algorithms. However, they cannot always guarantee to find the global optimum and may instead obtain the local optima. In this study, we propose a new method, the single FEA method, that can effectively and efficiently find the extremal load conditions of a given structure. The new method considers all possible arrangements of prescribed loads without needing to create and analyze the corresponding finite element models. This is achieved by utilizing a single finite element analysis (FEA) with multiple load cases, where each load case has a unit load applied at one of the candidate load locations. Using the proposed method, we can quickly obtain the extremal load arrangements of the structure to produce the best and worst stiffness performance. A variety of 2D and 3D examples are presented to demonstrate the effectiveness and wide applicability of the new method.

A spatio-temporal fully meshless method for hyperbolic PDEs

We introduce a meshless method derived by considering the time variable as a spatial variable without the need to extend further conditions to the solution of linear and non-linear hyperbolic PDEs. The method is based on the moving least squares method, more precisely in the Generalized Finite Difference Method which allows us to select well-conditioned stars. Several 2D and 3D examples including the time variable are shown for both regular and irregular node distributions. The results are compared with explicit GFDM both in terms of errors and execution time.

A continuous model for connectivity constraints in topology optimization

The aim of this work is to present a continuos mathematical model that characterizes and enforces connectivity in a topology optimization problem. That goal is accomplished by constraining the second eigenvalue of an auxiliary eigenproblem, solved together with the governing state law in each step of the iterative process. Our density-based approach is illustrated with 2d and 3d numerical examples in the context of structural design.

Topological design of cellular structures for maximum shear modulus using homogenization SEMDOT

This paper incorporates the homogenization theory into non-penalized Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT) algorithm to conduct the design of cellular structures with the maximum shear modulus. The parametric study and comparison with existing results obtained by BESO are carried out. The numerical examples in 2D and 3D demonstrate the effectiveness of non-penalized SEMDOT in generating smooth cellular structures with the maximum shear modulus. Compared to BESO, SEMDOT can achieve comparable results and smoother boundaries. Smooth boundaries obtained by SEMDOT can facilitate the manufacturing of obtained cellular structures in 3D or 4D printing.

Studying spin diffusion and quantum entanglement with LF-µSR

LF-µSR studies have previously been used to study the diffusive 1D motion of solitons and polarons in conducting polymers. This type of study was also applied to investigating the diffusive motion of spinons in spin-1/2 antiferromagnetic chains. Recently the method has been extended to examples of 2D layered triangular spin lattices which can support quantum spin liquid states, such as 1T-TaS2 and YbZnGaO4. These systems are found to show spin dynamics that matches well to 2D spin diffusion, such a model being found to provide a much better fit to the data than previously proposed models for spin correlations in such systems. In YbZnGaO4 the diffusion rate shows a clear crossover between classical and quantum regimes as T falls below the exchange coupling J. That the spin diffusion approach works well in the high T classical region might be expected, but it is found that it also works equally well in the low T quantum region where quantum entanglement controls the spin dynamics. Measurement of the diffusion rate allows a T dependent length scale to be derived from the data that can be assigned to a quantum entanglement length ξ E. Another entanglement measure, the Quantum Fisher Information F Q can also be obtained from the data and its T dependence is compared to that of ξ E.

Iterative geostatistical seismic inversion with rock-physics constraints for permeability prediction

Accurate prediction of the spatial distribution of subsurface permeability is a fundamental task in reservoir characterization and monitoring studies for hydrocarbon production and CO2geologic storage. Predicting permeability over large areas is challenging, due to their high variability and spatial anisotropy. Common approaches for modeling permeability generally involve deterministic calculations from porosity using precalibrated rock-physics models (RPMs) or geostatistical cosimulation methods that reproduce observed experimental porosity-permeability relationships. Instead, we have predicted permeability from seismic data using an iterative geostatistical seismic inversion method that combines the advantages of rock-physics and geostatistical modeling methods. First, we simulate facies through 1D vertical Markov chain simulations. Then, permeability, porosity, and acoustic impedance are sequentially generated and conditioned to the previously simulated facies model. An RPM is used to evaluate the misfit between the permeability predictions obtained from geostatistical cosimulation at the well locations and well-log values computed from the acoustic impedance. The residuals of the misfit function are used as conditioning constraints in the stochastic update of the models in the subsequent iteration. The outcome of our methodology is a set of multiple geostatistical realizations of facies, permeability, porosity, and acoustic impedance conditioned to seismic data and constrained by an RPM. We first illustrate the method on a synthetic 1D example and compare it to a traditional geostatistical inversion approach. We then apply our inversion to a 3D real data set to assess the methodology performance with scarce conditioning data and in the presence of noise.

A Reconstructed Method of Acoustic Logging Data and Its Application in Seismic Lithological Inversion for Uranium Reservoir

As a sedimentary mineral, most sandstone type uranium deposits are formed in petroliferous basins. Therefore, we can fully tap the residual economic value of historical logging and 3D seismic data measured for oil and gas to search for sandstone type uranium deposits. However, a large amount of acoustic logging data are missing in the target stratum of the uranium reservoir in that it is not the main stratum of oil and gas. A reconstructed method of acoustic logging data based on clustering analysis and with the low-frequency compensation of deterministic inversion is proposed to solve this problem. Secondly, we can use these logging data with seismic data to obtain the 3D inversion data volume representing the sand body of the uranium reservoir based on seismic lithological inversion. Then, we can also delimit the 3D spatial range of sandstone type uranium deposits in petroliferous basins based on the calibration of uranium anomaly and sub-body detection. Finally, a 3D field data example is given to test and analyze the effectiveness of the above research schemes.

Provably Positive Central Discontinuous Galerkin Schemes via Geometric Quasilinearization for Ideal MHD Equations

In the numerical simulation of ideal magnetohydrodynamics (MHD), keeping the pressure and density always positive is essential for both physical considerations and numerical stability. This is however a challenging task, due to the underlying relation between such a positivity-preserving (PP) property and the magnetic divergence-free (DF) constraint as well as the strong nonlinearity of the MHD equations. In this paper, we present the first rigorous PP analysis of the central discontinuous Galerkin (CDG) methods and construct arbitrarily high-order provably PP CDG schemes for ideal MHD. By the recently developed geometric quasilinearization (GQL) approach, our analysis reveals that the PP property of standard CDG methods is closely related to a discrete magnetic DF condition, whose form was unknown prior to our analysis and differs from that for the noncentral discontinuous Galerkin (DG) and finite volume methods in [K. Wu, SIAM J. Numer. Anal., 56 (2018), pp. 2124–2147]. The discovery of this relation lays the foundation for the design of our PP CDG schemes. In the one-dimensional (1D) case, the discrete DF condition is naturally satisfied, and we rigorously prove that the standard CDG method is PP under a condition that can be enforced easily with an existing PP limiter. However, in the multidimensional cases, the corresponding discrete DF condition is highly nontrivial, yet critical, and we analytically prove that the standard CDG method, even with the PP limiter, is not PP in general, as it generally fails to meet the discrete DF condition. We address this issue by carefully analyzing the structure of the discrete divergence terms and then constructing new locally DF CDG schemes for Godunov’s modified MHD equations with an additional source term. The key point is to find out the suitable discretization of the source term such that it exactly cancels out all the terms in the discovered discrete DF condition. Based on the GQL approach, we prove in theory the PP property of the new multidimensional CDG schemes under a CFL condition. The robustness and accuracy of the proposed PP CDG schemes are further validated by several demanding 1D, two-dimensional, and three-dimensional numerical MHD examples, including the high-speed jets and blast problems with very low plasma beta.

Reducing the number of different members in truss layout optimization

Despite the long history of the truss layout optimization approach, its practical applications have been limited, partly due to high manufacturing costs associated with complex optimized structures consisting of members with different cross-sectional areas and member lengths. To address this issue, this study considers optimizing truss structures comprising limited types of members. On this topic, two distinct problems are considered, wherein the first problem, members of the same type share the same cross-sectional area (i.e., section-type problem); and in the second problem, members of the same type share the same cross-sectional area and length (i.e., member-type problem). A novel post-processing approach is proposed to tackle the target problems. In this approach, the optimized structures from the traditional layout and geometry optimization approaches are used as the starting points, members of which are then separated into groups by the k-means clustering approach. Subsequently, the clustered structures are geometrically optimized to reduce the area and length deviations (i.e., the differences between member area/length values and the corresponding cluster means). Several 2D and 3D examples are presented to demonstrate the capability of the proposed approaches. For the section-type problem, the area deviations can be reduced to near 0 for any given cluster number. The member-type problem is relatively more complex, but by providing more clusters, the maximum length deviation can be reduced below the target thresholds. Through the proposed clustering approach, the number of different members in the optimized trusses can be substantially decreased, thereby significantly reducing manufacturing costs.

Learning Invariance Preserving Moment Closure Model for Boltzmann–BGK Equation

As one of the main governing equations in kinetic theory, the Boltzmann equation is widely utilized in aerospace, microscopic flow, etc. Its high-resolution simulation is crucial in these related areas. However, due to the high dimensionality of the Boltzmann equation, high-resolution simulations are often difficult to achieve numerically. The moment method which was first proposed in Grad (Commun Pure Appl Math 2(4):331–407, 1949) is among the popular numerical methods to achieve efficient high-resolution simulations. We can derive the governing equations in the moment method by taking moments on both sides of the Boltzmann equation, which effectively reduces the dimensionality of the problem. However, one of the main challenges is that it leads to an unclosed moment system, and closure is needed to obtain a closed moment system. It is truly an art in designing closures for moment systems and has been a significant research field in kinetic theory. Other than the traditional human designs of closures, the machine learning-based approach has attracted much attention lately in Han et al. (Proc Natl Acad Sci USA 116(44):21983–21991, 2019) and Huang et al. (J Non-Equilib Thermodyn 46(4):355–370, 2021). In this work, we propose a machine learning-based method to derive a moment closure model for the Boltzmann–BGK equation. In particular, the closure relation is approximated by a carefully designed deep neural network that possesses desirable physical invariances, i.e., the Galilean invariance, reflecting invariance, and scaling invariance, inherited from the original Boltzmann–BGK equation and playing an important role in the correct simulation of the Boltzmann equation. Numerical simulations on the 1D–1D examples including the smooth and discontinuous initial condition problems, Sod shock tube problem, the shock structure problems, and the 1D–3D examples including the smooth and discontinuous problems demonstrate satisfactory numerical performances of the proposed invariance preserving neural closure method.

Structurally constrained inversion by means of a Minimum Gradient Support regularizer: examples of FD-EMI data inversion constrained by GPR reflection data

SUMMARYMany geophysical inverse problems are known to be ill-posed and, thus, requiring some kind of regularization in order to provide a unique and stable solution. A possible approach to overcome the inversion ill-posedness consists in constraining the position of the model interfaces. For a grid-based parameterization, such a structurally constrained inversion can be implemented by adopting the usual smooth regularization scheme in which the local weight of the regularization is reduced where an interface is expected. By doing so, sharp contrasts are promoted at interface locations while standard smoothness constraints keep affecting the other regions of the model. In this work, we present a structurally constrained approach and test it on the inversion of frequency-domain electromagnetic induction (FD-EMI) data using a regularization approach based on the Minimum Gradient Support stabilizer, which is capable to promote sharp transitions everywhere in the model, i.e., also in areas where no structural a prioriinformation is available. Using 1D and 2D synthetic data examples, we compare the proposed approach to a structurally constrained smooth inversion as well as to more standard (i.e., not structurally constrained) smooth and sharp inversions. Our results demonstrate that the proposed approach helps in finding a better and more reliable reconstruction of the subsurface electrical conductivity distribution, including its structural characteristics. Furthermore, we demonstrate that it allows to promote sharp parameter variations in areas where no structural information are available. Lastly, we apply our structurally constrained scheme to FD-EMI field data collected at a field site in Eastern Germany to image the thickness of peat deposits along two selected profiles. In this field example, we use collocated constant offset ground-penetrating radar (GPR) data to derive structural a priori information to constrain the inversion of the FD-EMI data. The results of this case study demonstrate the effectiveness and flexibility of the proposed approach.

A New Surface Node Method to Accurately Model the Mechanical Behavior of the Boundary in 3D State-Based Peridynamics

Peridynamics is a non-local continuum theory capable of modeling crack initiation and propagation in solid bodies. However, the layer near the boundary of the body exhibits a stiffness fluctuation due to the so-called surface effect and the inaccurate way of imposing the boundary conditions. Moreover, in numerical models discretized using the meshfree method with uniform grid spacing, there are no nodes on the external surface of the body where the boundary conditions should be applied. Inspired by the method of the fictitious nodes with the Taylor-based extrapolation, we propose an innovative method that introduces a new type of nodes lying on the external surface of the body, i.e., the surface nodes. These nodes represent the interactions between the nodes within the body and the fictitious nodes surrounding the body, and they are used to mitigate the surface effect and properly impose the boundary conditions via the concept of force flux. Moreover, a procedure to compute the analytical solution of peridynamic problems is developed: a manufactured displacement field is prescribed and the volume and surface forces, to obtain that displacement field, are computed. The benefits of the surface node method are shown by means of several 2D and 3D quasi-static examples by comparing the numerical results with other methods with or without boundary corrections.

An online autonomous learning and prediction scheme for machine learning assisted structural optimization

In this paper, an online autonomous learning and prediction (ALP) scheme is proposed for machine learning assisted structural optimization (MLaSO) to reduce total computational time with controlled accuracy of objective function prediction. In MLaSO, information in selected routine iterations is used via online learning to train a Neural Network, which is then used to perform online prediction of information in subsequent other iterations. Although using MLaSO with pre-defined learning and prediction scheme can save total computational time, it is difficult to determine the most suitable learning and prediction scheme prior to solving an optimization problem. The present ALP method enables autonomous activation of online learnings and predictions via maximizing specific computational efficiency while constraining the accuracy of the objective function prediction. Four numerical 2D and 3D examples are presented to demonstrate the computational benefits of integrating ALP with gradient based MLaSO method. The present numerical results demonstrate that: (a) using ALP in MLaSO can save more computational time than using the pre-defined scheme with similar prediction accuracy, and (b) more computational time can be saved by allowing larger tolerance in the accuracy of objective function prediction.

Casting-oriented structural topology optimization with dimensional shrinkage

Solidification shrinkage and surface finishing cause the inevitable dimensional shrinkage between the blueprint and model designs, thereby creating huge cost and precision challenges to casting designs. This paper proposes an efficient structural topology optimization approach to simultaneously achieve the blueprint and model designs with uniform dimensional shrinkage in only one optimization setting. An appropriate two-projected-field scheme is established to simulate the blueprint and model designs by extending the density-based threshold projection approach. And, an effective quantitative framework is constructed to precisely predict the shrinkage rules. Besides, to ensure the castability of both the two projected designs, the Poisson equation-based constraint method is subtly integrated with the two-projected-field scheme. More importantly, to reduce the computational burden involving multiple designs and constraints, an efficient formulation is presented by reasonably considering the projected fields in the objective and constraint functions. Typical 2D and 3D numerical examples as well as one real-world example are provided to validate the effectiveness of the proposed strategies. The results show that the traditional one-projected-field scheme is difficult to yield the model design through the filtered field associated with the blueprint design. The proposed approach enables this and can help engineers achieve predictable dimensional shrinkage in an optimization way.