Problems In Computational Mechanics Research Articles

Interpolation-based immersogeometric analysis methods for multi-material and multi-physics problems

AbstractImmersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted meshes, immersed boundary methods instead embed the computational domain in a structured background grid. Interpolation-based immersed boundary methods augment existing finite element software to non-invasively implement immersed boundary capabilities through extraction. Extraction interpolates the structured background basis as a linear combination of Lagrange polynomials defined on a foreground mesh, creating an interpolated basis that can be easily integrated by existing methods. This work extends the interpolation-based immersed isogeometric method to multi-material and multi-physics problems. Beginning from level-set descriptions of domain geometries, Heaviside enrichment is implemented to accommodate discontinuities in state variable fields across material interfaces. Adaptive refinement with truncated hierarchically refined B-splines (THB-splines) is used to both improve interface geometry representations and to resolve large solution gradients near interfaces. Multi-physics problems typically involve coupled fields where each field has unique discretization requirements. This work presents a novel discretization method for coupled problems through the application of extraction, using a single foreground mesh for all fields. Numerical examples illustrate optimal convergence rates for this method in both 2D and 3D, for partial differential equations representing heat conduction, linear elasticity, and a coupled thermo-mechanical problem. The utility of this method is demonstrated through image-based analysis of a composite sample, where in addition to circumventing typical meshing difficulties, this method reduces the required degrees of freedom when compared to classical boundary-fitted finite element methods.

Formulating and heuristic solving of contact problems in hybrid data-driven computational mechanics

In this work we consider the hybrid Data-Driven Computational Mechanics (DDCM) approach, in which a smooth constitutive manifold is reconstructed to obtain a well-behaved nonlinear optimization problem (NLP) rather than the much harder discrete-continuous NLP (DCNLP) of the direct DDCM approach. The key focus is on the addition of geometric inequality constraints to the hybrid DDCM formulation. Therein, the required constraint force leads to a contact problem in the form of a mathematical program with complementarity constraints (MPCC), a problem class that is still less complex than the DCNLP. For this MPCC we propose a heuristic quick-shot solution approach, which can produce verifiable solutions by solving up to four NLPs. We perform various numerical experiments on three different contact problems of increasing difficulty to demonstrate the potential and limitations of this approach.

Taylor Polynomials in a High Arithmetic Precision as Universal Approximators

Function approximation is a fundamental process in a variety of problems in computational mechanics, structural engineering, as well as other domains that require the precise approximation of a phenomenon with an analytic function. This work demonstrates a unified approach to these techniques, utilizing partial sums of the Taylor series in a high arithmetic precision. In particular, the proposed approach is capable of interpolation, extrapolation, numerical differentiation, numerical integration, solution of ordinary and partial differential equations, and system identification. The method employs Taylor polynomials and hundreds of digits in the computations to obtain precise results. Interestingly, some well-known problems are found to arise in the calculation accuracy and not methodological inefficiencies, as would be expected. In particular, the approximation errors are precisely predictable, the Runge phenomenon is eliminated, and the extrapolation extent may a priory be anticipated. The attained polynomials offer a precise representation of the unknown system as well as its radius of convergence, which provides a rigorous estimation of the prediction ability. The approximation errors are comprehensively analyzed for a variety of calculation digits and test problems and can be reproduced by the provided computer code.

Hybrid methods for solving structural geometric nonlinear dynamic equations: Implementation of fifth-order iterative procedures within composite time integration methods

This paper studies a class of hybrid methods that implement multi-point iterative procedures as the nonlinear solver within optimized composite implicit methods. These multi-point methods are employed to enhance convergence order and accuracy without requiring higher-order derivatives for solving structural geometric nonlinear dynamic equations. The promising approach provides an opportunity to delve deeper into its implications and explore the potential applications of multi-point techniques and composite methods across the analysis of dynamical systems. This study involves the utilization of two-point and five-point iterative methods with fifth-order convergence within three-step and four-step optimized composite implicit time integration methods. Moreover, a comprehensive hybrid scheme is introduced by integrating a family of optimized composite implicit methods with single-point and multi-point iterative methods. Then, the performance of the hybrid scheme is analyzed through structural examples, considering factors such as time step size, tolerance threshold, and the degrees of freedom of structures. The findings and numerical results illustrate that multi-point methods outperform the single-point method in terms of the number of iterations. Simultaneously, the four-step time integration method exhibits superior performance compared to the three-step time integration method, albeit with a marginal difference. It is observed that the hybrid method, consisting of the four-step time integration method and the five-point iterative method, performs better than other methods in terms of the number of iterations and errors for solving geometric nonlinear dynamic equations, especially in large structures. Thus, it stands out as a favorable choice for practical analyses and a possible candidate for generalization to other types of nonlinear problems in computational mechanics.

A simple spectral representation of a second-order symmetric tensor and its variation

The spectral decomposition of a second-order, symmetric tensor is widely adopted in many fields of Computational Mechanics. As an example, in elasto-plasticity under large strain and rotations, given the Cauchy deformation tensor, it is a fundamental step to compute the logarithmic strain tensor.Recently, this approach has also been adopted in small-strain isotropic plasticity to reconstruct the stress tensor as a function of its eigenvalues, allowing the formulation of predictor–corrector return algorithms in the invariants space. These algorithms not only reduce the number of unknowns at the constitutive level, but also allow the correct handling of stress states in which the plastic normals are undefined, thus ensuring a better convergence with respect to the standard approach.While the eigenvalues of a symmetric, second-order tensor can be simply computed as a function of the tensor invariants, the computation of its eigenbasis can be more difficult, especially when two or more eigenvalues are coincident. Moreover, when a Newton–Raphson algorithm is adopted to solve nonlinear problems in Computational Mechanics, also the tensorial derivatives of the eigenbasis, whose computation is still more complicated, are required to assemble the tangent matrix.A simple and comprehensive method is presented, which can be adopted to compute a closed form representation of a second-order tensor, as well as their derivatives with respect to the tensor itself, allowing a simpler and numerically accurate implementation of spectral decomposition of a tensor in Computational Mechanics applications.

Voronoi discretization to improve the meshless local Petrov–Galerkin method in 3D-computational fracture mechanics

Purpose1) The OE-MLPG penalty meshfree method is developed to solve cracked structure.(2) Smartening the numerical meshfree method by combining the particle swarm optimization (PSO) optimization algorithms and Voronoi computational geometric algorithm. (3). Selection of base functions, finding optimal penalty factor and distribution of appropriate nodal points to the accuracy of calculation in the meshless local Petrov–Galekrin (MLPG) meshless method.Design/methodology/approachUsing appropriate shape functions and distribution of nodal points in local domains and sub-domains and choosing an approximation or interpolation method has an effective role in the application of meshless methods for the analysis of computational fracture mechanics problems, especially problems with geometric discontinuity and cracks. In this research, computational geometry technique, based on the Voronoi diagram (VD) and Delaunay triangulation and PSO algorithm, are used to distribute nodal points in the sub-domain of analysis (crack line and around it on the crack plane).FindingsBy doing this process, the problems caused by too closeness of nodal points in computationally sensitive areas that exist in general methods of nodal point distribution are also solved. Comparing the effect of the number of sentences of basic functions and their order in the definition of shape functions, performing the mono-objective PSO algorithm to find the penalty factor, the coefficient, convergence, arrangement of nodal points during the three stages of VD implementation and the accuracy of the answers found indicates, the efficiency of V-E-MLPG method with Ns = 7 and ß = 0.0037–0.0075 to estimation of 3D-stress intensity factors (3D-SIFs) in computational fracture mechanics.Originality/valueThe present manuscript is a continuation of the studies (Ref. [33]) carried out by the authors, about; feasibility assessment, improvement and solution of challenges, introduction of more capacities and capabilities of the numerical MLPG method have been used. In order to validate the modeling and accuracy of calculations, the results have been compared with the findings of reference article [34] and [35].

A Variational Formulation of Physics-Informed Neural Network for the Applications of Homogeneous and Heterogeneous Material Properties Identification

A new approach for solving computational mechanics problems using physics-informed neural networks (PINNs) is proposed. Variational forms of residuals for the governing equations of solid mechanics are utilized, and the residual is evaluated over the entire computational domain by employing domain decomposition and polynomials test functions. A parameter network is introduced and initial and boundary conditions, as well as data mismatch, are incorporated into a total loss function using a weighted summation. The accuracy of the model in solving forward problems of solid mechanics is demonstrated to be higher than that of the finite element method (FEM). Furthermore, homogeneous and heterogeneous material distributions can be effectively captured by the model using limited observations, such as strain components. This contribution is significant for potential applications in non-destructive evaluation, where obtaining detailed information about the material properties is difficult.

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 .

BINN: A deep learning approach for computational mechanics problems based on boundary integral equations

We proposed the boundary-integral type neural networks (BINN) for the boundary value problems in computational mechanics. The boundary integral equations are employed to transfer all the unknowns to the boundary, then the unknowns are approximated using neural networks and solved through a training process. The loss function is chosen as the residuals of the boundary integral equations. Regularization techniques are adopted to efficiently evaluate the weakly singular and Cauchy principal integrals in boundary integral equations. Potential problems and elastostatic problems are mainly concerned in this article as a demonstration. The proposed method has several outstanding advantages: First, the dimensions of the original problem are reduced by one, thus the degrees of freedom are greatly reduced. Second, the proposed method does not require any extra treatment to introduce the boundary conditions, since they are naturally considered through the boundary integral equations. Therefore, the method is suitable for complex geometries. Third, BINN is suitable for problems on the infinite or semi-infinite domains. Moreover, BINN can easily handle heterogeneous problems with a single neural network without domain decomposition.

MYCRUNCHGPT: A LLM ASSISTED FRAMEWORK FOR SCIENTIFIC MACHINE LEARNING

Scientific machine learning (SciML) has advanced recently across many different areas in computational science and engineering. The objective is to integrate data and physics seamlessly without the need of employing elaborate and computationally taxing data assimilation schemes. However, preprocessing, problem formulation, code generation, postprocessing, and analysis are still time- consuming and may prevent SciML from wide applicability in industrial applications and in digital twin frameworks. Here, we integrate the various stages of SciML under the umbrella of ChatGPT, to formulate MyCrunchGPT, which plays the role of a conductor orchestrating the entire workflow of SciML based on simple prompts by the user. Specifically, we present two examples that demonstrate the potential use of MyCrunchGPT in optimizing airfoils in aerodynamics, and in obtaining flow fields in various geometries in interactive mode, with emphasis on the validation stage. To demonstrate the flow of the MyCrunchGPT, and create an infrastructure that can facilitate a broader vision, we built a web app based guided user interface, that includes options for a comprehensive summary report. The overall objective is to extend MyCrunchGPT to handle diverse problems in computational mechanics, design, optimization and controls, and general scientific computing tasks involved in SciML, hence using it as a research assistant tool but also as an educational tool. While here the examples focus on fluid mechanics, future versions will target solid mechanics and materials science, geophysics, systems biology, and bioinformatics.

Evaluation of global sensitivity analysis methods for computational structural mechanics problems

Abstract The curse of dimensionality confounds the comprehensive evaluation of computational structural mechanics problems. Adequately capturing complex material behavior and interacting physics phenomenon in models can lead to long run times and memory requirements resulting in the need for substantial computational resources to analyze one scenario for a single set of input parameters. The computational requirements are then compounded when considering the number and range of input parameters spanning material properties, loading, boundary conditions, and model geometry that must be evaluated to characterize behavior, identify dominant parameters, perform uncertainty quantification, and optimize performance. To reduce model dimensionality, global sensitivity analysis (GSA) enables the identification of dominant input parameters for a specific structural performance output. However, many distinct types of GSA methods are available, presenting a challenge when selecting the optimal approach for a specific problem. While substantial documentation is available in the literature providing details on the methodology and derivation of GSA methods, application-based case studies focus on fields such as finance, chemistry, and environmental science. To inform the selection and implementation of a GSA method for structural mechanics problems for a nonexpert user, this article investigates five of the most widespread GSA methods with commonly used structural mechanics methods and models of varying dimensionality and complexity. It is concluded that all methods can identify the most dominant parameters, although with significantly different computational costs and quantitative capabilities. Therefore, method selection is dependent on computational resources, information required from the GSA, and available data.

The Class of Atomic Exponential Basis Functions EFupn(x,ω)-Development and Application

The purpose of this paper is to present the class of atomic basis functions (ABFs) which are of exponential type and are denoted by . While ABFs of the algebraic type are already represented in the numerical modeling of various problems in mathematical physics and computational mechanics, ABFs of the exponential type have not yet been sufficiently researched. These functions, unlike the ABFs of the algebraic type , contain the tension parameter , which gives them additional approximation properties. Exponential monomials up to the th degree can be described exactly by the linear combination of the functions . The function for is called the “mother” ABF of the exponential type, i.e., . In other words, the functions are elements of the linear vector space and retain all the properties of their “mother” function . Thus, this paper, in terms of its content and purpose, can be understood as a sequel of the article by Brajčić Kurbaša et al., which shows the basic properties and application of the basis function This paper presents, in an analogous way, the development and application of the exponential basis functions . Here, for the first time, expressions for calculating the values of the functions and their derivatives are given in a form suitable for application in numerical analyses, which is shown in the verification examples of the approximations of known functions.

Bayesian calibration of coupled computational mechanics models under uncertainty based on interface deformation

Calibration or parameter identification is used with computational mechanics models related to observed data of the modeled process to find model parameters such that good similarity between model prediction and observation is achieved. We present a Bayesian calibration approach for surface coupled problems in computational mechanics based on measured deformation of an interface when no displacement data of material points is available. The interpretation of such a calibration problem as a statistical inference problem, in contrast to deterministic model calibration, is computationally more robust and allows the analyst to find a posterior distribution over possible solutions rather than a single point estimate. The proposed framework also enables the consideration of unavoidable uncertainties that are present in every experiment and are expected to play an important role in the model calibration process. To mitigate the computational costs of expensive forward model evaluations, we propose to learn the log-likelihood function from a controllable amount of parallel simulation runs using Gaussian process regression. We introduce and specifically study the effect of three different discrepancy measures for deformed interfaces between reference data and simulation. We show that a statistically based discrepancy measure results in the most expressive posterior distribution. We further apply the approach to numerical examples in higher model parameter dimensions and interpret the resulting posterior under uncertainty. In the examples, we investigate coupled multi-physics models of fluid–structure interaction effects in biofilms and find that the model parameters affect the results in a coupled manner.

Deep energy method in topology optimization applications

This paper explores the possibilities of applying physics-informed neural networks (PINNs) in topology optimization (TO) by introducing a fully self-supervised TO framework based on PINNs. This framework solves the forward elasticity problem by the deep energy method (DEM). Instead of training a separate neural network to update the density distribution, we leverage the fact that the compliance minimization problem is self-adjoint to express the element sensitivity directly in terms of the displacement field from the DEM model. Thus, no additional neural network is needed for the inverse problem. The method of moving asymptotes is used as the optimizer for updating density distribution. The implementation of Neumann, Dirichlet, and periodic boundary conditions is described in the context of the DEM model. Three numerical examples are presented to demonstrate framework capabilities: (i) compliance minimization in 2D under different geometries and loading, (ii) compliance minimization in 3D, and (iii) maximization of homogenized shear modulus to design 2D metamaterial unit cells. The results show that the optimized designs from the DEM-based framework are very comparable to those generated by the finite element method and shed light on a new way of integrating PINN-based simulation methods into classical computational mechanics problems.

Wavelet Neural Operator for solving parametric partial differential equations in computational mechanics problems

With massive advancements in sensor technologies and Internet-of-things (IoT), we now have access to terabytes of historical data; however, there is a lack of clarity on how to best exploit the data to predict future events. One possible alternative in this context is to utilize an operator learning algorithm that directly learns the nonlinear mapping between two functional spaces; this facilitates real-time prediction of naturally arising complex evolutionary dynamics. In this work, we introduce a novel operator learning algorithm referred to as the Wavelet Neural Operator (WNO) that blends integral kernel with wavelet transformation. WNO harnesses the superiority of the wavelets in time–frequency localization of the functions and enables accurate tracking of patterns in the spatial domain and effective learning of the functional mappings. Since the wavelets are localized in both time/space and frequency, WNO can provide high spatial and frequency resolution. This offers learning of the finer details of the parametric dependencies in the solution for complex problems. The efficacy and robustness of the proposed WNO are illustrated on a wide array of problems involving Burger’s equation, Darcy flow, Navier–Stokes equation, Allen–Cahn equation, and Wave advection equation. A comparative study with respect to existing operator learning frameworks is presented. Finally, the proposed approach is used to build a digital twin capable of predicting Earth’s air temperature based on available historical data.

Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics

We present a new Integrated Finite Element Neural Network framework (I-FENN), with the objective to accelerate the numerical solution of nonlinear computational mechanics problems. We leverage the swift predictive capability of neural networks (NNs) and we embed them inside the finite element stiffness function, to compute element-level state variables and their derivatives within a nonlinear, iterative numerical solution. This process is conducted jointly with conventional finite element methods that involve shape functions: the NN receives input data that resembles the material point deformation and its output is used to construct element-level field variables such as the element Jacobian matrix and residual vector. Here we introduce I-FENN to the continuum damage analysis of quasi-brittle materials, and we establish a new non-local gradient-based damage framework which operates at the cost of a local damage approach. First, we develop a physics informed neural network (PINN) to resemble the non-local gradient model and then we train the neural network offline. The network learns to predict the non-local equivalent strain at each material point, as well as its derivative with respect to the local strain. Then, the PINN is integrated in the element stiffness definition and conducts the local to non-local strain transformation, whereas the two PINN outputs are used to construct the element Jacobian matrix and residual vector. This process is carried out within the nonlinear solver, until numerical convergence is achieved. The resulting method bears the computational cost of the conventional local damage approach, but ensures mesh-independent results and a diffused non-local strain and damage profile. As a result, the proposed method tackles the vital drawbacks of both the local and non-local gradient method, respectively being the mesh-dependence and additional computational cost. We showcase through a series of numerical examples the computational efficiency and generalization capability of I-FENN in the context of non-local continuum damage, and we discuss the future outlook. The PINN training code and associated training data files have been made publicly available online, to enable reproducibility of our results.

Interval deep learning for computational mechanics problems under input uncertainty

Solving problems in complex engineering systems often involves employing a cascade of computational models which solve different sub-problems. In computational mechanics in particular, complex systems that use model cascading under uncertainty include important safety-critical applications (e.g., aerospace and nuclear reactor systems). There are two fundamental considerations to solve problems with cascading models. First, the collaborating models involved in solving the problem must share compatible data formats and modeling assumptions. Second, the models must be capable of processing and propagating the uncertainty on the raw measurements and the uncertainty on the predictions from other models. The first consideration is often well-understood and implemented in practice. On the other hand, the second consideration, despite its importance, is less understood and implemented. We argue that in such complex engineering systems it is necessary to employ models capable of processing data uncertainty to obtain reliable predictions for the problem. One main difficulty in uncertainty quantification is that there is often not enough information about the data to make probabilistic assumptions. In fact, sometimes the only information available to the analyst is in the form of upper and lower bounds of the data. In such cases, interval analysis (IA) offers an alternative to quantify the uncertainty. Another difficulty is that we generally do not have direct access to the value of an uncertain mechanical quantity but only to indirect measurements or models that are used to represent it [1]. Indeed, the data uncertainty that flows through the model cascade can arise from measurements or from predictions of other models. To solve these problems, we present a novel interval deep neural network (DINN) that is capable of providing reliable input data to a mechanics model from data containing uncertainty. A numerical experiment is conducted using a dataset of concrete mix measurements with interval uncertainty to predict concrete strength. In general, the DINN can be used to solve multiple problems under uncertainty in science and engineering.

Ensemble Approximate Control Variate Estimators: Applications to MultiFidelity Importance Sampling

The recent growth in multifidelity uncertainty quantification has given rise to a large set of variance reduction techniques that leverage information from model ensembles to provide variance reduction for estimates of the statistics of a high-fidelity model. In this paper we provide two contributions: (1) we utilize an ensemble estimator to account for uncertainties in the optimal weights of approximate control variate (ACV) approaches and derive lower bounds on the number of samples required to guarantee variance reduction; and (2) we extend an existing multifidelity importance sampling (MFIS) scheme to leverage control variates. Our approach directly addresses a limitation of many multifidelity sampling strategies that require the usage of pilot samples to estimate covariances. As such we make significant progress towards both increasing the practicality of approximate control variates—for instance, by accounting for the effect of pilot samples—and using multifidelity approaches more effectively for estimating low-probability events. The numerical results indicate our hybrid MFIS-ACV estimator achieves up to 50% improvement in variance reduction over the existing state-of-the-art MFIS estimator, which had already shown an outstanding convergence rate compared to the Monte Carlo method, on several problems of computational mechanics.

Elasto-capillary fluid–structure interaction with compound droplets

Computational modeling of elasto-capillarity, i.e., a fluid–structure interaction phenomenon where the solid deformation is driven by capillary forces at fluid–fluid interfaces has recently emerged as an important problem in computational mechanics. However, the high-fidelity simulation of elasto-capillary problems involving three immiscible fluids has remained unexplored. Here, we present a mathematical model and an algorithm to simulate elasto-capillary problems involving compound droplets. Such problems are of pivotal importance in pharmaceutical, biological and food industrial applications. We adopt a phase-field model described by the ternary Navier–Stokes–Cahn–Hilliard equations for the three immiscible fluids and a neo-Hookean model for the solid. We define a fluid–solid surface energy function, which determines the fluid–solid wettability and the tractions transmitted to the solid at the fluid–solid interface. We adopt a boundary-fitted approach for our fluid–structure interaction formulation and Isogeometric Analysis for the spatial discretization. To demonstrate the effectiveness of our computational model and algorithm, we perform elasto-capillary simulations involving various types of compound droplets.

Preface

XXII International conference on Computational Mechanics and Modern Applied Software Systems CMMASS 2021 September 4-13, 2021 Preface This issue comprises selected works presented on the 22nd International Conference on Computational Mechanics and Modern Applied Software Systems (CMMASS 2021) held from 4 September to 13 September 2021 in Alushta (The Republic of Crimea). The Conference was hosted by Moscow Aviation Institute (national research university) in cooperation with the Laboratory «Space flight mechanics» of the Moscow Aviation Institute, the Russian Academy of Sciences, the Ministry of Science and Higher Education of the Russian Federation, the Russian National Committee on Theoretical and Applied Mechanics.The objective of the Conference is to raise and discuss up-to-date questions relating to computational mechanics problems, modern software systems, parallel computing problems, far-reaching numerical approaches and various aspects of application of computational tools to solve problems in mechanics of a deformable solid, fluid dynamics, aerospace mechanics, nanomechanics and other fields. Also during the Conference the problems of software and database support with analysis of instruments for data processing and data visualization and new achievements from researches are considered.The Conference became a workspace for participants from Russia, the People’s Republic of China, Japan, Australia, India, Poland, Serbia, France, Italy, Armenia, Republic of Kazakhstan.List of Conference Committee, Co-Chairs, Deputy Chairman and Program Committee are available in this pdf.