Decomposition for Parallel Computing: Graph Theory Methods

This chapter starts with presenting simple and general methods for calculating the degree of static indeterminacy of different types of skeletal structures, such as rigid-jointed planar and space frames, pin-jointed planar trusses and ball-jointed space trusses. Then the progress made in the force method of structural analysis in recent years is presented. Efficient methods are developed for the formation of highly sparse flexibility matrices for different types of skeletal structures.

Most of researches for large-scale parallel structural analysis using finite element method have focused on iterative solution methods since direct solution methods generally have many difficulties and disadvantages for large-scale problems. However, due to the numerical robustness of direct methods that guarantees the solution to be obtained within estimated time, direct methods are much more desirable for general application of large-scale structural analysis, as well as due to ease of use which makes most of structural engineers prefer direct methods. In this research, we propose a parallel direct solver for large-scale finite element analysis that can overcome most of difficulties and disadvantages of direct methods by using domain-wise approach for direct methods. By using the proposed solver with our own structural analysis code, we can show good scalability as a direct method and can solve the largest problem ever solved by direct solvers.

Singularity conditions that arise during structural optimization can seriously degrade the performance of the optimizer. The singularities are intrinsic to the formulation of the structural optimization problem and are not associated with the method of analysis. Certain conditions that give rise to singularities have been identified in earlier papers, encompassing the entire structure. Further examination revealed more complex sets of conditions in which singularities occur. Some of these singularities are local in nature, being associated with only a segment of the structure. Moreover, the likelihood that one of these local singularities may arise during an optimization procedure can be much greater than that of the global singularity identified earlier. Examples are provided of these additional forms of singularities. A framework is also given in which these singularities can be recognized. In particular, the singularities can be identified by examination of the stress displacement relations along with the compatibility conditions and/or the displacement stress relations derived in the integrated force method of structural analysis.

The goal of this work is to predict the effect of part geometry and process parameters on the instantaneous spatiotemporal distribution of temperature, also called the thermal field or temperature history, in metal parts as they are being built layer-by-layer using additive manufacturing (AM) processes. In pursuit of this goal, the objective of this work is to develop and verify a graph theory-based approach for predicting the temperature distribution in metal AM parts. This objective is consequential to overcome the current poor process consistency and part quality in AM. One of the main reasons for poor part quality in metal AM processes is ascribed to the nature of temperature distribution in the part. For instance, steep thermal gradients created in the part during printing leads to defects, such as warping and thermal stress-induced cracking. Existing nonproprietary approaches to predict the temperature distribution in AM parts predominantly use mesh-based finite element analyses that are computationally tortuous—the simulation of a few layers typically requires several hours, if not days. Hence, to alleviate these challenges in metal AM processes, there is a need for efficient computational models to predict the temperature distribution, and thereby guide part design and selection of process parameters instead of expensive empirical testing. Compared with finite element analyses techniques, the proposed mesh-free graph theory-based approach facilitates prediction of the temperature distribution within a few minutes on a desktop computer. To explore these assertions, we conducted the following two studies: (1) comparing the heat diffusion trends predicted using the graph theory approach with finite element analysis, and analytical heat transfer calculations based on Green’s functions for an elementary cuboid geometry which is subjected to an impulse heat input in a certain part of its volume and (2) simulating the laser powder bed fusion metal AM of three-part geometries with (a) Goldak’s moving heat source finite element method, (b) the proposed graph theory approach, and (c) further comparing the thermal trends predicted from the last two approaches with a commercial solution. From the first study, we report that the thermal trends approximated by the graph theory approach are found to be accurate within 5% of the Green’s functions-based analytical solution (in terms of the symmetric mean absolute percentage error). Results from the second study show that the thermal trends predicted for the AM parts using graph theory approach agree with finite element analyses, and the computational time for predicting the temperature distribution was significantly reduced with graph theory. For instance, for one of the AM part geometries studied, the temperature trends were predicted in less than 18 min within 10% error using the graph theory approach compared with over 180 min with finite element analyses. Although this paper is restricted to theoretical development and verification of the graph theory approach, our forthcoming research will focus on experimental validation through in-process thermal measurements.

With the development of computer hardware, the size of finite element analysis is getting larger and larger. Still, it is practically impossible to perform structural analysis of large scale structure with only one analysis. Structural analysis of large scale structures like ship is usually carried out on the models of several different scale level, varying from whole ship, cargo hold parts and to the detail structures. Currently, the finite element models for different analysis level is generated from different geometric model, mainly because the unit geometry of mesh generation is different in each level of the analysis. In this study, a prototype system is developed that automatically generates finite element models for different analysis level from single ship geometry data. (See Fig. 1)

Bipolar disorder is often misdiagnosed as major depressive disorder, which leads to inadequate treatment. Depressed individuals versus healthy control subjects, show increased expectation of negative outcomes. Due to increased impulsivity and risk for mania, however, depressed individuals with bipolar disorder may differ from those with major depressive disorder in neural mechanisms underlying anticipation processes. Graph theory methods for neuroimaging data analysis allow the identification of connectivity between multiple brain regions without prior model specification, and may help to identify neurobiological markers differentiating these disorders, thereby facilitating development of better therapeutic interventions. This study aimed to compare brain connectivity among regions involved in win/loss anticipation in depressed individuals with bipolar disorder (BDD) versus depressed individuals with major depressive disorder (MDD) versus healthy control subjects using graph theory methods. The study was conducted at the University of Pittsburgh Medical Center and included 31 BDD, 39 MDD, and 36 healthy control subjects. Participants were scanned while performing a number guessing reward task that included the periods of win and loss anticipation. We first identified the anticipatory network across all 106 participants by contrasting brain activation during all anticipation periods (win anticipation + loss anticipation) versus baseline, and win anticipation versus loss anticipation. Brain connectivity within the identified network was determined using the Independent Multiple sample Greedy Equivalence Search (IMaGES) and Linear non-Gaussian Orientation, Fixed Structure (LOFS) algorithms. Density of connections (the number of connections in the network), path length, and the global connectivity direction ('top-down' versus 'bottom-up') were compared across groups (BDD/MDD/healthy control subjects) and conditions (win/loss anticipation). These analyses showed that loss anticipation was characterized by denser top-down fronto-striatal and fronto-parietal connectivity in healthy control subjects, by bottom-up striatal-frontal connectivity in MDD, and by sparse connectivity lacking fronto-striatal connections in BDD. Win anticipation was characterized by dense connectivity of medial frontal with striatal and lateral frontal cortical regions in BDD, by sparser bottom-up striatum-medial frontal cortex connectivity in MDD, and by sparse connectivity in healthy control subjects. In summary, this is the first study to demonstrate that BDD and MDD with comparable levels of current depression differed from each other and healthy control subjects in density of connections, connectivity path length, and connectivity direction as a function of win or loss anticipation. These findings suggest that different neurobiological mechanisms may underlie aberrant anticipation processes in BDD and MDD, and that distinct therapeutic strategies may be required for these individuals to improve coping strategies during expectation of positive and negative outcomes.

Summary form only given. Time-domain electric field integral equations (TD-EFIE) are widely used for analyzing transient scattering from perfect electrically conducting surfaces. These equations often are solved via marching-on-in-time (MOT) methods. Unfortunately, the computational cost of classically formulated MOT-based TD-EFIE solvers scales as O(N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) where N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> and N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> are the number of time steps and spatial surface unknowns, respectively. This cost prevents these solvers from being used to study high-frequency phenomena involving large-scale structures.In the past, a variety of methods for accelerating MOT-based TD-EFIE solvers have been developed; they include, but are not limited to, hybrid TD-EFIE and time domain physical optics (TDPO) methods, and fast summation schemes such as the time-domain adaptive integral method (TD-AIM). In the hybrid TD-EFIE-PO method, structures are partitioned into EFIE and PO regions. In the EFIE region, surface currents are updated by MOT; incident fields comprise those produced by external sources and currents in the PO region. In the PO region, currents are updated using incident fields produced by external sources and currents in the EFIE region (W. Luo, W. Y. Yin, M. D. Zhu, J. Y. Zhao, and J. F. Mao, IEEE Trans. Electromagn. Compat., 54, 1006-1016, 2012). The asymptotic computational complexity of the TD-EFIE-PO scheme scales like that of the TD-EFIE scheme albeit with a far smaller leading constant, enabling the solution of much larger problems. The TD-AIM constitutes the extension of the frequency domain adaptive integral method to the time domain, and reduces the computational complexity of a TD-EFIE solver from O(NtNs ) to 2 O ( N t N s log N s ) without any (high-frequency) 3/2 2 assumptions on the current (A. E. Yılmaz, J. M. Jin, and E. Michielssen, IEEE Trans. Antennas Propag., 52, 2692-2708, 2004). This paper presents a solver that hybridizes the TD-EFIE-PO and TD-AIM methods. In the hybrid scheme, the TD-AIM is used to propagate fields within the EFIE region as well as from the EFIE to the PO region, and back. The hybrid scheme is computationally far more efficient than its standalone constituents. Depending on the nature of the structure considered, the TD-AIM method uses a single grid that covers both the EFIE and PO regions, or separate and nonoverlapping grids for both regions. We apply the proposed scheme to the analysis of interactions of high-power electromagnetic pulses (HP-EMP) with large-scale platforms. Numerical results demonstrate the efficiency, accuracy, and stability of the proposed method.

Fault location of tribo-mechanical systems—a graph theory and matrix approach

In response to the requirement, the manufacturing processes have evolved from mass production technique, through its implementation on reconfigurable systems. The most difficult part of reconfigurable manufacturing systems (RMSs) is an understanding of the barriers related to their relative interdependencies. This article presents a conceptual framework of various barriers associated with RMSs along with their interdependencies. Using graph theory and matrix methods, the barriers of RMSs presented in the form of a single numerical index. The proposed model is quite versatile from the point of view that it provides an opportunity to integrate new barriers which could impact on manufacturing processes. In the present work, an attempt has been made to develop a mathematical model of these barriers using graph theory and matrix methods. An index of barriers in RMS is proposed which evaluates the inhibiting power of these barriers.

In recent years, the educational landscape has witnessed a transformative shift with the emergence of game-based learning (GBL) as a dynamic and effective strategy. This innovative educational approach engages students in a captivating manner, fostering the acquisition of essential skills vital for their intellectual growth. Among these skills are problem-solving, critical thinking, and algorithmic reasoning, which play pivotal roles in preparing students for the challenges of the modern world. This research article takes a pioneering step by delving into the integration of graph theory concepts into educational games. This approach is conceived as a novel method to amplify cognitive skills, offering a unique blend of theoretical knowledge and practical application. Graph theory, renowned for its abstraction and representation of relationships between entities, brings a fresh dimension to the educational gaming experience, promising enhanced learning outcomes for students. The core objective of this study is to meticulously explore the impact of graph-based educational games on students' overall learning outcomes. Furthermore, the research endeavors to assess how these immersive games contribute to the development and application of algorithmic thinking in real-world scenarios. By seamlessly integrating graph theory into the fabric of educational games, this study aims to uncover the potential synergies between engaging gameplay and the cultivation of critical cognitive skills.

Objective To explore the changes in brain structure network connection in children with attention deficit hyperactivity disorder(ADHD), and to provide novel markers for early identification of ADHD in clinical practice. Methods Deterministic diffusion-tensor tractography and graph theory approaches were used to investigate the topologic organization of the brain structural connectome in 25 children with ADHD and 23 healthy control children from May 2017 to May 2018, at Children′s Hospital of Xuzhou Medical University.Individual white matter networks were constructed for each participant, then the global properties, nodal properties and edge-wise distributions were compared between the two groups. Results (1)The global efficiency of the ADHD group (0.30±0.13) was significantly lower than that of the healthy control group (0.38±0.11), but the clustering coefficient (0.35±0.28) and the characteristic path length (2.94±0.38) were significantly higher than those of the healthy control group (0.28±0.10, 2.65±0.37), and the differences were statistically significant (t=-2.41, 2.31, 2.62, all P<0.05). (2)In the ADHD group, the nodal efficiency of the left inferior frontal gyrus, triangular part (0.13±0.06), left supramarginal gyrus (0.30±0.10), left inferior parietal, angular gyri (0.29±0.10), left precuneus (0.26±0.12)were significantly lower than the healthy control group(0.17±0.07, 0.38±0.10, 0.40±0.12, 0.35±0.12), while the nodal efficiency of the right superior frontal gyrus, orbital part and right paracentral lobule were significantly higher than the healthy control group(0.49±0.17, 0.43±0.14), and the differences were statistically significant[t=-2.52, -2.62, -3.11, -2.77, 2.34, 2.79, all P<0.05, false discovery rate(FDR) corrected]. (3)A disrupted subnetwork was observed that consisted of left frontoparietal areas, basal ganglia, thalamus and insular network (P<0.05, FDR corrected), which has the potential to discriminate individuals with ADHD from healthy control children(area under receiver operating characteristic curve was 0.78). (4)Diminished strength of the subnet work connections was correlated with the attention defect in patients with ADHD(r=-0.607, P=0.003). Conclusions Using magnetic resonance diffusion tensor imaging, with the help of graph theory analysis technology, ADHD children can be observed changes in brain structure network at multiple levels.The distribution pattern of brain network structure connection changes is expected to become a new marker for identifying ADHD. Key words: Attention deficit hyperactivity disorder; Diffusion tensor image; Graph theory; Structural network

This paper presents the selection of cutting parameters of heavy cutting process in side milling using graph theory and matrix approach (GTMA). The multiple attribute decision-making methods are also applied to rank and select the cutting parameters for the given application. The objective is to find the optimized cutting parameters in side milling operation. The considered performance characteristics are tool wear rate and metal removal rate. In GTMA approach, a “performance suitability index (PSI)” evaluates and ranks the performance characteristics from the given alternatives. It is registered that the performance, for which the value of PSI is highest, is the best choice for the given application. The index is derived from the graph of performance selection factors. An example is included to illustrate the approach.

The purpose of this paper is to investigate the intensity of variables of the flexible manufacturing system (FMS) which affect flexibility in FMS. The study was performed by conducting a cross–sectional survey within manufacturing firms in India through exploratory factor analysis (EFA) and graph theory and matrix approach (GTMA). By performing EFA, factor structure is identified whereas GTMA quantified the factor structure of a set of observed variables. GTMA is a multiple attribute decision making (MADM) methodology used to find the intensity of variables in an organisation. In this paper, EFA is applied to extract the factors in FMS by The Statistical Package for Social Sciences (SPSS 20) software and evaluation of intensity of variables affecting flexibility in FMS by GTMA. Fifteen variables are identified through literature, and four factors extracted, which involves the flexibility of FMS in production flexibility, machine flexibility, product flexibility, and volume flexibility.

A material selection model using graph theory and matrix approach

&lt;p&gt;The analysis of the new football stadium roof raised a bunch of analytical problems mainly connected with the large volume of information to be processed rather than the scale of the structure itself. The paper discusses some of those problems, methods of their solution as well as some tricks aimed at decreasing the amount of data and improving the analytical performance for the structural analysis of large scale structures.&lt;/p&gt;