Stéphane P. A. Bordas

Vinh Phu Nguyen, Stéphane P. A. Bordas, Timon Rabczuk

Isogeometric analysis (IGA) represents a recently developed technology in computational mechanics that offers the possibility of integrating methods for analysis and Computer Aided Design (CAD) into a single, unified process. The implications to practical engineering design scenarios are profound, since the time taken from design to analysis is greatly reduced, leading to dramatic gains in efficiency. The tight coupling of CAD and analysis within IGA requires knowledge from both fields and it is one of the goals of the present paper to outline much of the commonly used notation. In this manuscript, through a clear and simple Matlab implementation, we present an introduction to IGA applied to the Finite Element (FE) method and related computer implementation aspects. Furthermore, implemen- tation of the extended IGA which incorporates enrichment functions through the partition of unity method (PUM) is also presented, where several examples for both two-dimensional and three-dimensional fracture are illustrated. The open source Matlab code which accompanies the present paper can be applied to one, two and three-dimensional problems for linear elasticity, linear elastic fracture mechanics, structural mechanics (beams/plates/shells including large displacements and rotations) and Poisson problems with or without enrichment. The Bezier extraction concept that allows FE analysis to be performed efficiently on T-spline geometries is also incorporated. The article includes a summary of recent trends and developments within the field of IGA.

Lucca Schek, Peter Lewintan, Wolfgang Müller et al.

We introduce a new method, dubbed Geometric Structure-Preserving Interpolation ($Γ$-SPIN) to preserve physics-constraints inherent in the material parameter limits of the finite-strain Cosserat micropolar model. The method advocates to interpolate the Cosserat rotation tensor using geodesic elements, which maintain objectivity and correctly represent curvature measures. At the same time, it proposes relaxing the interaction between the rotation tensor and the deformation tensor to alleviate locking effects. This relaxation is achieved in two steps. First, the regularity of the Cosserat rotation tensor is reduced by interpolating it into the Nédélec space. Second, the resulting field is projected back onto the Lie-group of rotations. Together, these steps define a lower-regularity projection-based interpolation. The construction allows the discrete Cosserat rotation tensor to match the polar part of the discrete deformation tensor. This ensures stable behaviour in the asymptotic regime as the Cosserat couple modulus tends to infinity, which constrains the model towards its couple-stress limit. We establish the consistency, stability, and optimality of the proposed method through several benchmark problems. The study culminates in a demonstration of its efficacy on a more intricate curved domain, contrasted with outcomes obtained from conventional interpolation techniques.

Octavio A. González-Estrada, Sundararajan Natarajan, Juan José Ródenas et al.

An error control technique aimed to assess the quality of smoothed finite element approximations is presented in this paper. Finite element techniques based on strain smoothing appeared in 2007 were shown to provide significant advantages compared to conventional finite element approximations. In particular, a widely cited strength of such methods is improved accuracy for the same computational cost. Yet, few attempts have been made to directly assess the quality of the results obtained during the simulation by evaluating an estimate of the discretization error. Here we propose a recovery type error estimator based on an enhanced recovery technique. The salient features of the recovery are: enforcement of local equilibrium and, for singular problems a "smooth+singular" decomposition of the recovered stress. We evaluate the proposed estimator on a number of test cases from linear elastic structural mechanics and obtain precise error estimations whose effectivities, both at local and global levels, are improved compared to recovery procedures not implementing these features.

Saurabh Deshpande, Stéphane P. A. Bordas, Jakub Lengiewicz

In many cutting-edge applications, high-fidelity computational models prove to be too slow for practical use and are therefore replaced by much faster surrogate models. Recently, deep learning techniques have increasingly been utilized to accelerate such predictions. To enable learning on large-dimensional and complex data, specific neural network architectures have been developed, including convolutional and graph neural networks. In this work, we present a novel encoder-decoder geometric deep learning framework called MAgNET, which extends the well-known convolutional neural networks to accommodate arbitrary graph-structured data. MAgNET consists of innovative Multichannel Aggregation (MAg) layers and graph pooling/unpooling layers, forming a graph U-Net architecture that is analogous to convolutional U-Nets. We demonstrate the predictive capabilities of MAgNET in surrogate modeling for non-linear finite element simulations in the mechanics of solids.

Huu Phuoc Bui, Satyendra Tomar, Stéphane P. A. Bordas

This paper describes the use of the corotational cut Finite Element Method (FEM) for real-time surgical simulation. Users only need to provide a background mesh which is not necessarily conforming to the boundaries/interfaces of the simulated object. The details of the surface, which can be directly obtained from binary images, are taken into account by a multilevel embedding algorithm applied to elements of the background mesh that cut by the surface. Boundary conditions can be implicitly imposed on the surface using Lagrange multipliers. The implementation is verified by convergence studies with optimal rates. The algorithm is applied to various needle insertion simulations (e.g. for biopsy or brachytherapy) into brain and liver to verify the reliability of method, and numerical results show that the present method can make the discretisation independent from geometric description, and can avoid the complexity of mesh generation of complex geometries while retaining the accuracy of the standard FEM. Using the proposed approach is very suitable for real-time and patient specific simulations as it improves the simulation accuracy by taking into account automatically and properly the simulated geometry.

Octavio A. González-Estrada, Juan José Ródenas, Stéphane P. A. Bordas et al.

Purpose: This paper aims at assessing the effect of (1) the statical admissibility of the recovered solution; (2) the ability of the recovered solution to represent the singular solution; on the accuracy, local and global effectivity of recovery-based error estimators for enriched finite element methods (e.g. the extended finite element method, XFEM). Design/methodology/approach: We study the performance of two recovery techniques. The first is a recently developed superconvergent patch recovery procedure with equilibration and enrichment (SPR-CX). The second is known as the extended moving least squares recovery (XMLS), which enriches the recovered solutions but does not enforce equilibrium constraints. Both are extended recovery techniques as the polynomial basis used in the recovery process is enriched with singular terms for a better description of the singular nature of the solution. Findings: Numerical results comparing the convergence and the effectivity index of both techniques with those obtained without the enrichment enhancement clearly show the need for the use of extended recovery techniques in Zienkiewicz-Zhu type error estimators for this class of problems. The results also reveal significant improvements in the effectivities yielded by statically admissible recovered solutions. Originality/value: This work shows that both extended recovery procedures and statical admissibility are key to an accurate assessment of the quality of enriched finite element approximations.

Saurabh Deshpande, Raúl I. Sosa, Stéphane P. A. Bordas et al.

Deep learning surrogate models are being increasingly used in accelerating scientific simulations as a replacement for costly conventional numerical techniques. However, their use remains a significant challenge when dealing with real-world complex examples. In this work, we demonstrate three types of neural network architectures for efficient learning of highly non-linear deformations of solid bodies. The first two architectures are based on the recently proposed CNN U-NET and MAgNET (graph U-NET) frameworks which have shown promising performance for learning on mesh-based data. The third architecture is Perceiver IO, a very recent architecture that belongs to the family of attention-based neural networks--a class that has revolutionised diverse engineering fields and is still unexplored in computational mechanics. We study and compare the performance of all three networks on two benchmark examples, and show their capabilities to accurately predict the non-linear mechanical responses of soft bodies.

Alejandro Ortiz-Bernardin, Philip Köbrich, Jack S. Hale et al.

We introduce a novel meshfree Galerkin method for the solution of Reissner-Mindlin plate problems that is written in terms of the primitive variables only (i.e., rotations and transverse displacement) and is devoid of shear-locking. The proposed approach uses linear maximum-entropy approximations and is built variationally on a two-field potential energy functional wherein the shear strain, written in terms of the primitive variables, is computed via a volume-averaged nodal projection operator that is constructed from the Kirchhoff constraint of the three-field mixed weak form. The stability of the method is rendered by adding bubble-like enrichment to the rotation degrees of freedom. Some benchmark problems are presented to demonstrate the accuracy and performance of the proposed method for a wide range of plate thicknesses.

Xujun Zhao, Stéphane P. A. Bordas, Jianmin Qu

Interfacial energy plays an important role in equilibrium morphologies of nanosized microstructures of solid materials due to the high interface-to-volume ratio, and can no longer be neglected as it does in conventional mechanics analysis. The present work develops an effective numerical approach by means of a hybrid smoothed extended finite element/level set method to model nanoscale inhomogeneities with interfacial energy effect, in which the finite element mesh can be completely independent of the interface geometry. The Gurtin-Murdoch surface elasticity model is used to account for the interface stress effect and the Wachspress interpolants are used for the first time to construct the shape functions in the smoothed extended finite element method. Selected numerical results are presented to study the accuracy and efficiency of the proposed method as well as the equilibrium shapes of misfit particles in elastic solids. The presented results compare very well with those obtained from theoretical solutions and experimental observations, and the computational efficiency of the method is shown to be superior to that of its most advanced competitor.

Eleni D. Koronaki, Geremy Loachamin Suntaxi, Paris Papavasileiou et al.

Important variables of processes are often categorical, i.e. names or labels representing, e.g. categories of inputs, or types of reactors or a sequence of steps. In this work, we use Natural Language Processing Models to derive embeddings of such inputs that represent their actual meaning, or reflect the "distances" between categories, i.e. how similar or dissimilar they are. This is a marked difference from the current standard practice of using binary, or one-hot encoding to replace categorical variables with sequences of ones and zeros. Combined with dimensionality reduction techniques, either linear such as Principal Component Analysis, or nonlinear such as Uniform Manifold Approximation and Projection, the proposed approach leads to a meaningful, low-dimensional feature space. The significance of obtaining meaningful embeddings is illustrated in the context of an industrial coating process for cutting tools that includes both numerical and categorical inputs. In this industrial process, subject matter expertise suggests that the categorical inputs are critical for determining the final outcome but this cannot be taken into account with the current state-of-the-art. The proposed approach enables feature importance which is a marked improvement compared to the current state-of-the-art in the encoding of categorical variables. The proposed approach is not limited to the case-study presented here and is suitable for applications with similar mix of categorical and numerical critical inputs.

Paris Papavasileiou, Dimitrios G. Giovanis, Gabriele Pozzetti et al.

This study introduces a machine learning framework tailored to large-scale industrial processes characterized by a plethora of numerical and categorical inputs. The framework aims to (i) discern critical parameters influencing the output and (ii) generate accurate out-of-sample qualitative and quantitative predictions of production outcomes. Specifically, we address the pivotal question of the significance of each input in shaping the process outcome, using an industrial Chemical Vapor Deposition (CVD) process as an example. The initial objective involves merging subject matter expertise and clustering techniques exclusively on the process output, here, coating thickness measurements at various positions in the reactor. This approach identifies groups of production runs that share similar qualitative characteristics, such as film mean thickness and standard deviation. In particular, the differences of the outcomes represented by the different clusters can be attributed to differences in specific inputs, indicating that these inputs are critical for the production outcome. Leveraging this insight, we subsequently implement supervised classification and regression methods using the identified critical process inputs. The proposed methodology proves to be valuable in scenarios with a multitude of inputs and insufficient data for the direct application of deep learning techniques, providing meaningful insights into the underlying processes.

Zhaoxiang Shen, Raúl I. Sosa, Jakub Lengiewicz et al.

Accurate prediction of many-body dispersion (MBD) interactions is essential for understanding the van der Waals forces that govern the behavior of many complex molecular systems. However, the high computational cost of MBD calculations limits their direct application in large-scale simulations. In this work, we introduce a machine learning surrogate model specifically designed to predict MBD forces in polymer melts, a system that demands accurate MBD description and offers structural advantages for machine learning approaches. Our model is based on a trimmed SchNet architecture that selectively retains the most relevant atomic connections and incorporates trainable radial basis functions for geometric encoding. We validate our surrogate model on datasets from polyethylene, polypropylene, and polyvinyl chloride melts, demonstrating high predictive accuracy and robust generalization across diverse polymer systems. In addition, the model captures key physical features, such as the characteristic decay behavior of MBD interactions, providing valuable insights for optimizing cutoff strategies. Characterized by high computational efficiency, our surrogate model enables practical incorporation of MBD effects into large-scale molecular simulations.

Elisa Gómez de Lope, Saurabh Deshpande, Ramón Viñas Torné et al.

Omics data analysis is crucial for studying complex diseases, but its high dimensionality and heterogeneity challenge classical statistical and machine learning methods. Graph neural networks have emerged as promising alternatives, yet the optimal strategies for their design and optimization in real-world biomedical challenges remain unclear. This study evaluates various graph representation learning models for case-control classification using high-throughput biological data from Parkinson's disease and control samples. We compare topologies derived from sample similarity networks and molecular interaction networks, including protein-protein and metabolite-metabolite interactions (PPI, MMI). Graph Convolutional Network (GCNs), Chebyshev spectral graph convolution (ChebyNet), and Graph Attention Network (GAT), are evaluated alongside advanced architectures like graph transformers, the graph U-net, and simpler models like multilayer perceptron (MLP). These models are systematically applied to transcriptomics and metabolomics data independently. Our comparative analysis highlights the benefits and limitations of various architectures in extracting patterns from omics data, paving the way for more accurate and interpretable models in biomedical research.

Geremy Loachamín Suntaxi, Paris Papavasileiou, Eleni D. Koronaki et al.

This work introduces a comprehensive approach utilizing data-driven methods to elucidate the deposition process regimes in Chemical Vapor Deposition (CVD) reactors and the interplay of physical mechanism that dominate in each one of them. Through this work, we address three key objectives. Firstly, our methodology relies on process outcomes, derived by a detailed CFD model, to identify clusters of "outcomes" corresponding to distinct process regimes, wherein the relative influence of input variables undergoes notable shifts. This phenomenon is experimentally validated through Arrhenius plot analysis, affirming the efficacy of our approach. Secondly, we demonstrate the development of an efficient surrogate model, based on Polynomial Chaos Expansion (PCE), that maintains accuracy, facilitating streamlined computational analyses. Finally, as a result of PCE, sensitivity analysis is made possible by means of Sobol' indices, that quantify the impact of process inputs across identified regimes. The insights gained from our analysis contribute to the formulation of hypotheses regarding phenomena occurring beyond the transition regime. Notably, the significance of temperature even in the diffusion-limited regime, as evidenced by the Arrhenius plot, suggests activation of gas phase reactions at elevated temperatures. Importantly, our proposed methods yield insights that align with experimental observations and theoretical principles, aiding decision-making in process design and optimization. By circumventing the need for costly and time-consuming experiments, our approach offers a pragmatic pathway towards enhanced process efficiency. Moreover, this study underscores the potential of data-driven computational methods for innovating reactor design paradigms.

Saurabh Deshpande, Jakub Lengiewicz, Stéphane P. A. Bordas

For many novel applications, such as patient-specific computer-aided surgery, conventional solution techniques of the underlying nonlinear problems are usually computationally too expensive and are lacking information about how certain can we be about their predictions. In the present work, we propose a highly efficient deep-learning surrogate framework that is able to accurately predict the response of bodies undergoing large deformations in real-time. The surrogate model has a convolutional neural network architecture, called U-Net, which is trained with force-displacement data obtained with the finite element method. We propose deterministic and probabilistic versions of the framework. The probabilistic framework utilizes the Variational Bayes Inference approach and is able to capture all the uncertainties present in the data as well as in the deep-learning model. Based on several benchmark examples, we show the predictive capabilities of the framework and discuss its possible limitations

Anja K. Leist, Matthias Klee, Jung Hyun Kim et al.

The uptake of machine learning (ML) approaches in the social and health sciences has been rather slow, and research using ML for social and health research questions remains fragmented. This may be due to the separate development of research in the computational/data versus social and health sciences as well as a lack of accessible overviews and adequate training in ML techniques for non data science researchers. This paper provides a meta-mapping of research questions in the social and health sciences to appropriate ML approaches, by incorporating the necessary requirements to statistical analysis in these disciplines. We map the established classification into description, prediction, and causal inference to common research goals, such as estimating prevalence of adverse health or social outcomes, predicting the risk of an event, and identifying risk factors or causes of adverse outcomes. This meta-mapping aims at overcoming disciplinary barriers and starting a fluid dialogue between researchers from the social and health sciences and methodologically trained researchers. Such mapping may also help to fully exploit the benefits of ML while considering domain-specific aspects relevant to the social and health sciences, and hopefully contribute to the acceleration of the uptake of ML applications to advance both basic and applied social and health sciences research.

Vasilis Krokos, Viet Bui Xuan, Stéphane P. A. Bordas et al.

Multiscale computational modelling is challenging due to the high computational cost of direct numerical simulation by finite elements. To address this issue, concurrent multiscale methods use the solution of cheaper macroscale surrogates as boundary conditions to microscale sliding windows. The microscale problems remain a numerically challenging operation both in terms of implementation and cost. In this work we propose to replace the local microscale solution by an Encoder-Decoder Convolutional Neural Network that will generate fine-scale stress corrections to coarse predictions around unresolved microscale features, without prior parametrisation of local microscale problems. We deploy a Bayesian approach providing credible intervals to evaluate the uncertainty of the predictions, which is then used to investigate the merits of a selective learning framework. We will demonstrate the capability of the approach to predict equivalent stress fields in porous structures using linearised and finite strain elasticity theories.

Benoît Piranda, Paweł Chodkiewicz, Paweł Hołobut et al.

We present a distributed framework for predicting whether a planned reconfiguration step of a modular robot will mechanically overload the structure, causing it to break or lose stability under its own weight. The algorithm is executed by the modular robot itself and based on a distributed iterative solution of mechanical equilibrium equations derived from a simplified model of the robot. The model treats inter-modular connections as beams and assumes no-sliding contact between the modules and the ground. We also provide a procedure for simplified instability detection. The algorithm is verified in the Programmable Matter simulator VisibleSim, and in real-life experiments on the modular robotic system Blinky Blocks.

Sundararajan Natarajan, Stéphane P. A. Bordas, Ean Tat Ooi

We revisit the cell-based smoothed finite element method (SFEM) for quadrilateral elements and extend it to arbitrary polygons and polyhedrons in 2D and 3D, respectively. We highlight the similarity between the SFEM and the virtual element method (VEM). Based on the VEM, we propose a new stabilization approach to the SFEM when applied to arbitrary polygons and polyhedrons. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. Later, the SFEM is combined with the scaled boundary finite element method to problems involving singularity within the framework of the linear elastic fracture mechanics in 2D.