Graph-based Representation Research Articles

Redefining ABA+ Semantics via Abstract Set-to-Set Attacks

Assumption-based argumentation (ABA) is a powerful defeasible reasoning formalism which is based on the interplay of assumptions, their contraries, and inference rules. ABA with preferences (ABA+) generalizes the basic model by allowing qualitative comparison between assumptions. The integration of preferences however comes with a cost. In ABA+, the evaluation under two central and well-established semantics---grounded and complete semantics---is not guaranteed to yield an outcome. Moreover, while ABA frameworks without preferences allow for a graph-based representation in Dung-style frameworks, an according instantiation for general ABA+ frameworks has not been established so far. In this work, we tackle both issues: First, we develop a novel abstract argumentation formalism based on set-to-set attacks. We show that our so-called Hyper Argumentation Frameworks (HYPAFs) capture ABA+. Second, we propose relaxed variants of complete and grounded semantics for HYPAFs that yield an extension for all frameworks by design, while still faithfully generalizing the established semantics of Dung-style Argumentation Frameworks. We exploit the newly established correspondence between ABA+ and HYPAFs to obtain variants for grounded and complete ABA+ semantics that are guaranteed to yield an outcome. Finally, we discuss basic properties and provide a complexity analysis. Along the way, we settle the computational complexity of several ABA+ semantics.

Schematics Retrieval Using Whole-Graph Embedding Similarity

This paper addresses the pressing environmental concern of plastic waste, particularly in the biopharmaceutical production sector, where single-use assemblies (SUAs) significantly contribute to this issue. To address and mitigate this problem, we propose a unique approach centered around the standardization and optimization of SUA drawings through digitization and structured representation. Leveraging the non-Euclidean properties of SUA drawings, we employ a graph-based representation, utilizing graph convolutional networks (GCNs) to capture complex structural relationships. Introducing a novel weakly supervised method for the similarity-based retrieval of SUA graph networks, we optimize graph embeddings in a low-dimensional Euclidean space. Our method demonstrates effectiveness in retrieving similar graphs that share the same functionality, offering a promising solution to reduce plastic waste in pharmaceutical assembly processes.

Speech emotion recognition via graph-based representations

Speech emotion recognition (SER) has gained an increased interest during the last decades as part of enriched affective computing. As a consequence, a variety of engineering approaches have been developed addressing the challenge of the SER problem, exploiting different features, learning algorithms, and datasets. In this paper, we propose the application of the graph theory for classifying emotionally-colored speech signals. Graph theory provides tools for extracting statistical as well as structural information from any time series. We propose to use the mentioned information as a novel feature set. Furthermore, we suggest setting a unique feature-based identity for each emotion belonging to each speaker. The emotion classification is performed by a Random Forest classifier in a Leave-One-Speaker-Out Cross Validation (LOSO-CV) scheme. The proposed method is compared with two state-of-the-art approaches involving well known hand-crafted features as well as deep learning architectures operating on mel-spectrograms. Experimental results on three datasets, EMODB (German, acted) and AESDD (Greek, acted), and DEMoS (Italian, in-the-wild), reveal that our proposed method outperforms the comparative methods in these datasets. Specifically, we observe an average UAR increase of almost 18%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$18\\%$$\\end{document}, 8%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8\\%$$\\end{document} and 13%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$13\\%$$\\end{document}, respectively.

Enhancing Urban Data Analysis: Leveraging Graph-Based Convolutional Neural Networks for a Visual Semantic Decision Support System.

The persistent increase in the magnitude of urban data, combined with the broad range of sensors from which it derives in modern urban environments, poses issues including data integration, visualization, and optimal utilization. The successful selection of suitable locations for predetermined commercial activities and public utility services or the reuse of existing infrastructure arise as urban planning challenges to be addressed with the aid of the aforementioned data. In our previous work, we have integrated a multitude of publicly available real-world urban data in a visual semantic decision support environment, encompassing map-based data visualization with a visual query interface, while employing and comparing several classifiers for the selection of appropriate locations for establishing parking facilities. In the current work, we challenge the best representative of the previous approach, i.e., random forests, with convolutional neural networks (CNNs) in combination with a graph-based representation of the urban input data, relying on the same dataset to ensure comparability of the results. This approach has been inspired by the inherent visual nature of urban data and the increased capability of CNNs to classify image-based data. The experimental results reveal an improvement in several performance indices, implying a promising potential for this specific combination in decision support for urban planning problems.

Drug–target affinity prediction with extended graph learning-convolutional networks

BackgroundHigh-performance computing plays a pivotal role in computer-aided drug design, a field that holds significant promise in pharmaceutical research. The prediction of drug–target affinity (DTA) is a crucial stage in this process, potentially accelerating drug development through rapid and extensive preliminary compound screening, while also minimizing resource utilization and costs. Recently, the incorporation of deep learning into DTA prediction and the enhancement of its accuracy have emerged as key areas of interest in the research community. Drugs and targets can be characterized through various methods, including structure-based, sequence-based, and graph-based representations. Despite the progress in structure and sequence-based techniques, they tend to provide limited feature information. Conversely, graph-based approaches have risen to prominence, attracting considerable attention for their comprehensive data representation capabilities. Recent studies have focused on constructing protein and drug molecular graphs using sequences and SMILES, subsequently deriving representations through graph neural networks. However, these graph-based approaches are limited by the use of a fixed adjacent matrix of protein and drug molecular graphs for graph convolution. This limitation restricts the learning of comprehensive feature representations from intricate compound and protein structures, consequently impeding the full potential of graph-based feature representation in DTA prediction. This, in turn, significantly impacts the models’ generalization capabilities in the complex realm of drug discovery.ResultsTo tackle these challenges, we introduce GLCN-DTA, a model specifically designed for proficiency in DTA tasks. GLCN-DTA innovatively integrates a graph learning module into the existing graph architecture. This module is designed to learn a soft adjacent matrix, which effectively and efficiently refines the contextual structure of protein and drug molecular graphs. This advancement allows for learning richer structural information from protein and drug molecular graphs via graph convolution, specifically tailored for DTA tasks, compared to the conventional fixed adjacent matrix approach. A series of experiments have been conducted to validate the efficacy of the proposed GLCN-DTA method across diverse scenarios. The results demonstrate that GLCN-DTA possesses advantages in terms of robustness and high accuracy.ConclusionsThe proposed GLCN-DTA model enhances DTA prediction performance by introducing a novel framework that synergizes graph learning operations with graph convolution operations, thereby achieving richer representations. GLCN-DTA does not distinguish between different protein classifications, including structurally ordered and intrinsically disordered proteins, focusing instead on improving feature representation. Therefore, its applicability scope may be more effective in scenarios involving structurally ordered proteins, while potentially being limited in contexts with intrinsically disordered proteins.

Traceable Group-Wise Self-Optimizing Feature Transformation Learning: A Dual Optimization Perspective

Feature transformation aims to reconstruct an effective representation space by mathematically refining the existing features. It serves as a pivotal approach to combat the curse of dimensionality, enhance model generalization, mitigate data sparsity, and extend the applicability of classical models. Existing research predominantly focuses on domain knowledge-based feature engineering or learning latent representations. However, these methods, while insightful, lack full automation and fail to yield a traceable and optimal representation space. An indispensable question arises: Can we concurrently address these limitations when reconstructing a feature space for a machine learning task? Our initial work took a pioneering step towards this challenge by introducing a novel self-optimizing framework. This framework leverages the power of three cascading reinforced agents to automatically select candidate features and operations for generating improved feature transformation combinations. Despite the impressive strides made, there was room for enhancing its effectiveness and generalization capability. In this extended journal version, we advance our initial work from two distinct yet interconnected perspectives: 1) We propose a refinement of the original framework, which integrates a graph-based state representation method to capture the feature interactions more effectively and develop different Q-learning strategies to alleviate Q-value overestimation further. 2) We utilize a new optimization technique (actor-critic) to train the entire self-optimizing framework in order to accelerate the model convergence and improve the feature transformation performance. Finally, to validate the improved effectiveness and generalization capability of our framework, we perform extensive experiments and conduct comprehensive analyses. These provide empirical evidence of the strides made in this journal version over the initial work, solidifying our framework’s standing as a substantial contribution to the field of automated feature transformation. To improve the reproducibility, we have released the associated code and data by the Github link https://github.com/coco11563/TKDD2023_code.

Mixtures Recomposition by Neural Nets: A Multidisciplinary Overview.

Artificial Neural Networks (ANNs) are transforming how we understand chemical mixtures, providing an expressive view of the chemical space and multiscale processes. Their hybridization with physical knowledge can bridge the gap between predictivity and understanding of the underlying processes. This overview explores recent progress in ANNs, particularly their potential in the 'recomposition' of chemical mixtures. Graph-based representations reveal patterns among mixture components, and deep learning models excel in capturing complexity and symmetries when compared to traditional Quantitative Structure-Property Relationship models. Key components, such as Hamiltonian networks and convolution operations, play a central role in representing multiscale mixtures. The integration of ANNs with Chemical Reaction Networks and Physics-Informed Neural Networks for inverse chemical kinetic problems is also examined. The combination of sensors with ANNs shows promise in optical and biomimetic applications. A common ground is identified in the context of statistical physics, where ANN-based methods iteratively adapt their models by blending their initial states with training data. The concept of mixture recomposition unveils a reciprocal inspiration between ANNs and reactive mixtures, highlighting learning behaviors influenced by the training environment.

Sentiment analysis of tweets using text and graph multi-views learning

With the surge of deep learning framework, various studies have attempted to address the challenges of sentiment analysis of tweets (data sparsity, under-specificity, noise, and multilingual content) through text and network-based representation learning approaches. However, limited studies on combining the benefits of textual and structural (graph) representations for sentiment analysis of tweets have been carried out. This study proposes a multi-view learning framework (end-to-end and ensemble-based) that leverages both text-based and graph-based representation learning approaches to enrich the tweet representation for sentiment classification. The efficacy of the proposed framework is evaluated over three datasets using suitable baseline counterparts. From various experimental studies, it is observed that combining both textual and structural views can achieve better performance of sentiment classification tasks than its counterparts.

Modeling a domain wall network in BiFeO3 with stochastic geometry and entropy-based similarity measure

A compact and tractable two-dimensional model to generate the topological network structure of domain walls in BiFeO3 thin films is presented in this study. Our method combines the stochastic geometry parametric model of the centroidal Voronoi tessellation optimized using the von Neumann entropy, a novel information-theoretic tool for networks. The former permits the generation of image-based stochastic artificial samples of domain wall networks, from which the network structure is subsequently extracted and converted to the graph-based representation. The von Neumann entropy, which reflects information diffusion across multiple spatiotemporal scales in heterogeneous networks, plays a central role in defining a fitness function. It allows the use of the network as a whole rather than using a subset of network descriptors to search for optimal model parameters. The optimization of the parameters is carried out by a genetic algorithm through the maximization of the fitness function and results in the desired graph-based network connectivity structure. Ground truth empirical networks are defined, and a dataset of network connectivity structures of domain walls in BiFeO3 thin films is undertaken through manual annotation. Both a versatile tool for manual network annotation of noisy images and a new automatic network extraction method for high-quality images are developed.

Explaining graph convolutional network predictions for clinicians-An explainable AI approach to Alzheimer's disease classification.

Graph-based representations are becoming more common in the medical domain, where each node defines a patient, and the edges signify associations between patients, relating individuals with disease and symptoms in a node classification task. In this study, a Graph Convolutional Networks (GCN) model was utilized to capture differences in neurocognitive, genetic, and brain atrophy patterns that can predict cognitive status, ranging from Normal Cognition (NC) to Mild Cognitive Impairment (MCI) and Alzheimer's Disease (AD), on the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. Elucidating model predictions is vital in medical applications to promote clinical adoption and establish physician trust. Therefore, we introduce a decomposition-based explanation method for individual patient classification. Our method involves analyzing the output variations resulting from decomposing input values, which allows us to determine the degree of impact on the prediction. Through this process, we gain insight into how each feature from various modalities, both at the individual and group levels, contributes to the diagnostic result. Given that graph data contains critical information in edges, we studied relational data by silencing all the edges of a particular class, thereby obtaining explanations at the neighborhood level. Our functional evaluation showed that the explanations remain stable with minor changes in input values, specifically for edge weights exceeding 0.80. Additionally, our comparative analysis against SHAP values yielded comparable results with significantly reduced computational time. To further validate the model's explanations, we conducted a survey study with 11 domain experts. The majority (71%) of the responses confirmed the correctness of the explanations, with a rating of above six on a 10-point scale for the understandability of the explanations. Strategies to overcome perceived limitations, such as the GCN's overreliance on demographic information, were discussed to facilitate future adoption into clinical practice and gain clinicians' trust as a diagnostic decision support system.

Graph-Represented Distribution Similarity Index for Full-Reference Image Quality Assessment.

In this paper, we propose a graph-represented image distribution similarity (GRIDS) index for full-reference (FR) image quality assessment (IQA), which can measure the perceptual distance between distorted and reference images by assessing the disparities between their distribution patterns under a graph-based representation. First, we transform the input image into a graph-based representation, which is proven to be a versatile and effective choice for capturing visual perception features. This is achieved through the automatic generation of a vision graph from the given image content, leading to holistic perceptual associations for irregular image regions. Second, to reflect the perceived image distribution, we decompose the undirected graph into cliques and then calculate the product of the potential functions for the cliques to obtain the joint probability distribution of the undirected graph. Finally, we compare the distances between the graph feature distributions of the distorted and reference images at different stages; thus, we combine the distortion distribution measurements derived from different graph model depths to determine the perceived quality of the distorted images. The empirical results obtained from an extensive array of experiments underscore the competitive nature of our proposed method, which achieves performance on par with that of the state-of-the-art methods, demonstrating its exceptional predictive accuracy and ability to maintain consistent and monotonic behaviour in image quality prediction tasks. The source code is publicly available at the following website https://github.com/Land5cape/GRIDS.

Wave2Graph: Integrating spectral features and correlations for graph-based learning in sound waves

This paper investigates a novel graph-based representation of sound waves inspired by the physical phenomenon of correlated vibrations. We propose a Wave2Graph framework for integrating multiple acoustic representations, including the spectrum of frequencies and correlations, into various neural computing architectures to achieve new state-of-the-art performances in sound classification. The capability and reliability of our end-to-end framework are evidently demonstrated in voice pathology for low-cost and non-invasive mass-screening of medical conditions, including respiratory illnesses and Alzheimer’s Dementia. We conduct extensive experiments on multiple public benchmark datasets (ICBHI and ADReSSo) and our real-world dataset (IJSound: Respiratory disease detection using coughs and breaths). Wave2Graph framework consistently outperforms previous state-of-the-art methods with a large magnitude, up to 7.65% improvement, promising the usefulness of graph-based representation in signal processing and machine learning.

Multi-Stage Network with Geometric Semantic Attention for Two-View Correspondence Learning.

The removal of outliers is crucial for establishing correspondence between two images. However, when the proportion of outliers reaches nearly 90%, the task becomes highly challenging. Existing methods face limitations in effectively utilizing geometric transformation consistency (GTC) information and incorporating geometric semantic neighboring information. To address these challenges, we propose a Multi-Stage Geometric Semantic Attention (MSGSA) network. The MSGSA network consists of three key modules: the multi-branch (MB) module, the GTC module, and the geometric semantic attention (GSA) module. The MB module, structured with a multi-branch design, facilitates diverse and robust spatial transformations. The GTC module captures transformation consistency information from the preceding stage. The GSA module categorizes input based on the prior stage's output, enabling efficient extraction of geometric semantic information through a graph-based representation and inter-category information interaction using Transformer. Extensive experiments on the YFCC100M and SUN3D datasets demonstrate that MSGSA outperforms current state-of-the-art methods in outlier removal and camera pose estimation, particularly in scenarios with a high prevalence of outliers. Source code is available at https://github.com/shuyuanlin.

Synergistic Detection of Multimodal Fake News Leveraging TextGCN and Vision Transformer

In today's digital age, the rapid spread of fake news is a pressing concern. Fake news, whether intentional or inadvertent, manipulates public sentiment and threatens the integrity of online information. To address this, effective detection and prevention methods are vital. Detecting and addressing this multimodal fake news is an intricate challenge as, unlike traditional news articles that predominantly rely on textual content, multimodal fake news leverages the persuasive power of visual elements, making its identification a formidable task. Manipulated images can significantly sway individuals' perceptions and beliefs, making the detection of such deceptive content complex. Our research introduces an innovative approach to multimodal fake news identification by presenting a fusion-based methodology that harnesses the capabilities of Text Graph Convolutional Neural Networks (TextGCN) and Vision Transformers (ViT) to effectively utilise both text and image modalities. The proposed Methodology starts with preprocessing textual content using TextGCN, allowing for the capture of intricate structural dependencies among words and phrases. Simultaneously, visual features are extracted from associated images using ViT. Through a fusion mechanism, these modalities seamlessly integrate, yielding superior embeddings. The primary contributions encompass an in-depth exploration of multimodal fake news detection through a fusion-based approach. What sets our approach apart from existing techniques is its integration of graph-based feature extraction through TextGCN. While previous methods predominantly rely on text or image features, our approach harnesses the additional semantic information and intricate relationships within a graph structure, in addition to image embeddings. This enables our method to capture more comprehensive understanding of the data, resulting in increased accuracy and reliability. Our experiments demonstrate the exceptional performance of our fusion-based approach, which leverages multiple modalities and incorporates graph-based representations and semantic relationships. This method outperformed single modalities of text or image, achieving an impressive accuracy of 94.17% using a neural network after fusion. By seamlessly integrating graph-based representations and semantic relationships, our fusion-based technique represents a significant stride in addressing the challenges posed by multimodal fake news.

Transformer-Based Graph Neural Networks for Outfit Generation

Suggesting complementary clothing items to compose an outfit is a process of emerging interest, yet it involves a fine understanding of fashion trends and visual aesthetics. Previous works have mainly focused on recommendation by scoring visual appeal and representing garments as ordered sequences or as collections of pairwise-compatible items. This limits the full usage of relations among clothes. We attempt to bridge the gap between outfit recommendation and generation by leveraging a graph-based representation of items in a collection. The work carried out in this paper, tries to build a bridge between outfit recommendation and generation, by discovering new appealing outfits starting from a collection of pre-existing ones. We propose a transformer-based architecture, named TGNN, which exploits multi-headed self attention to capture relations between clothing items in a graph as a message passing step in Convolutional Graph Neural Networks. Specifically, starting from a seed, i.e. one or more garments, outfit generation is performed by iteratively choosing the garment that is most compatible with the previously chosen ones. Extensive experimentations are conducted with two different datasets, demonstrating the capability of the model to perform seeded outfit generation as well as obtaining state of the art results on compatibility estimation tasks.

On the computation of Delaunay triangulations via genetic algorithms

In this work, we introduce a new approach for computing Delaunay triangulations. Delaunay triangulations have numerous applications in geolocation, communications and other ICT systems or practical applications. Having available various types of approaches for computing such structures is rather desired from an implementation and computational point of view. We adopt Genetic Algorithms for computing Delaunay triangulations and present the design and evaluation of our novel approach. We consider a set of points in the plane as vertices and connect them with edges, creating the point graph. We have developed in C++ an application framework based on genetic algorithms, called Delaunay_Genetic, which produces the Delaunay triangulation structure of a given set of points in the plane. Delaunay_Genetic considers a novel graph-based chromosome representation of desired solutions, creates an initial population of individuals (chromosomes), an initial generation, and produces from the original population (generation) new generations of individuals in each repetition of the genetic process of Reproduction. Each new generation emerges more robust than the previous one. Our evaluations have revealed that the Delaunay triangulation yielded by Delaunay_Genetic, achieves an accuracy of 98–100% of the optimal Delaunay triangulation, while maintaining good convergence speed. Despite its limitations in computational time and space, the proposed novel approach exhibits several complementary benefits to computational geometry based approaches, such as allowing the insertion of new points in the triangulation dynamically, leading to seamless adaptation to new conditions, parallelization of the computational process and tolerance to noise regarding the coordinates of the points. Therefore, this work provides a useful alternative approach for computing Delaunay triangulations.

Income estimation based on human mobility patterns and machine learning models

Sustainable and inclusive urban development requires a thorough understanding of income distribution and poverty. Recent related research has extensively explored the use of automatically generated sensor data to proxy economic activities. Notably, human mobility patterns have been found to exhibit strong associations with socioeconomic attributes and great potential for income estimation. However, the representation of complex human mobility patterns and their effectiveness in income estimation needs further investigation. To address this, we propose three representations of human mobility: mobility indicators, activity footprints, and travel graphs. These representations feed into various models, including XGBoost, a traditional machine learning model, a convolutional neural network (CNN), and a time-series graph neural network (GCRN). By leveraging public transit data from Shenzhen, our study demonstrates that graph-based representations and deep learning models outperform other approaches in income estimation. They excel in minimising information loss and handling complex data structures. Spatial contextual attributes, such as transport accessibility, are the most influential factors, while indicators related to activity extent, temporal rhythm, and intensity contribute comparatively less. In summary, this study highlights the potential of cutting-edge artificial intelligence tools and emerging human mobility data as an alternative approach to estimating income distribution and addressing poverty-related concerns.

Integrating Knowledge Graphs into Distribution Grid Decision Support Systems

Distribution grids are complex networks containing multiple pieces of equipment. These components are interconnected, and each of them is described by various attributes. A knowledge graph is an interesting data format that represents pieces of information as nodes and relations between the pieces as edges. In this paper, we describe the proposed vocabulary used to build a distribution system knowledge graph. We identify the concepts used in such graphs and a set of relations to represent links between concepts. Both provide a semantically rich representation of a system. Additionally, we offer a few illustrative examples of how a distributed system knowledge graph can be utilized to gain more insight into the operations of the grid. We show a simplified analysis of how outages can influence customers based on their locations and how adding DERs can influence/change it. These demonstrative use cases show that the graph-based representation of a distribution grid allows for integrating information of different types and how such a repository can be efficiently utilized. Based on the experiments with distribution system knowledge graphs presented in this article, we postulate that graph-based representation enables a novel way of storing information about power grids and facilitates interactive methods for their visualization and analysis.

Is fragment-based graph a better graph-based molecular representation for drug design? A comparison study of graph-based models

Graph Neural Networks (GNNs) have gained significant traction in various sectors of AI-driven drug design. Over recent years, the integration of fragmentation concepts into GNNs has emerged as a potent strategy to augment the efficacy of molecular generative models. Nonetheless, challenges such as symmetry breaking and potential misrepresentation of intricate cycles and undefined functional groups raise questions about the superiority of fragment-based graph representation over traditional methods. In our research, we undertook a rigorous evaluation, contrasting the predictive prowess of eight models—developed using deep learning algorithms—across 12 benchmark datasets that span a range of properties. These models encompass established methods like GCN, AttentiveFP, and D-MPNN, as well as innovative fragment-based representation techniques. Our results indicate that fragment-based methodologies, notably PharmHGT, significantly improve model performance and interpretability, particularly in scenarios characterized by limited data availability. However, in situations with extensive training, fragment-based molecular graph representations may not necessarily eclipse traditional methods. In summation, we posit that the integration of fragmentation, as an avant-garde technique in drug design, harbors considerable promise for the future of AI-enhanced drug design.

Graph-based representation of history-dependent material response in the Data-Driven Computational Mechanics framework

This work develops an extension of the Data-Driven Computational Mechanics approach introduced by Kirchdoerfer and Ortiz (2016) to history-dependent behaviour. The material database is replaced by a material directed graph, where vertices represent strain–stress pairs and arcs encode thermomechanically admissible transitions between them. A graph search algorithm is used to select local material databases representative of the incremental material behaviour. This study thus gives the mathematical tools for a structured data-driven representation of the material behaviour, designed to minimise the increase of the problem dimensionality. The performances of the proposed approach applied to elastoplasticity are studied on a single-element structure and a larger truss structure. A two-step solver is proposed to address numerical accuracy issues in the latter case.