Showing 9 results for Graph Theory
I. Karimi, M. S. Masoudi,
Volume 14, Issue 1 (1-2024)
Abstract
The main part of finite element analysis via the force method involves the formation of a suitable null basis for the equilibrium matrix. For an optimal analysis, the chosen null basis matrices should exhibit sparsity and banding, aligning with the characteristics of sparse, banded, and well-conditioned flexibility matrices. In this paper, an effective method is developed for the formation of null bases of finite element models (FEMs) consisting of shell elements. This leads to highly sparse and banded flexibility matrices. This is achieved by associating specific graphs to the FEM and choosing suitable subgraphs to generate the self-equilibrating systems (SESs) on these subgraphs. The effectiveness of the present method is showcased through two examples.
M. Shahrouzi,
Volume 14, Issue 2 (2-2024)
Abstract
During the process of continuum topology optimization some pattern discontinuities may arise. It is an important challenge to overcome such irregularities in order to achieve or interpret the true optimal layout. The present work offers an efficient algorithm based on graph theoretical approach regarding density priorities. The developed method can recognize and handle solid continuous regions in a pre-optimized media. An illustrative example shows how the proposed priority guided trees can successfully distinguish the most crucial parts of the continuum during topology optimization.
R. Sheikholeslami, A. Kaveh,
Volume 15, Issue 1 (1-2025)
Abstract
The stability of large complex systems is a fundamental question in various scientific disciplines, from natural ecosystems to engineered environmental networks. This paper examines the interplay between network complexity and stability through the lens of graph theory and spectral analysis, based on Robert May’s seminal work on stability in randomly connected networks. Environmental systems are modeled as graphs in which components, such as reservoirs in a water distribution system or physical processes in hydrological cycle, interact through defined connections of varying strengths. Stability in these networks depends on the level of connectivity, the number of interacting components, and the strength of interactions between them. Previous studies have shown that as a system becomes more interconnected, it reaches a threshold beyond which it transitions sharply from stability to instability. Using concepts from spectral graph theory, we show how structural properties of an environmental network—such as degree distribution, modularity, and spectral characteristics—shape stability. Two numerical examples are presented to illustrate how increasing connectivity affects stability in water resource networks modeled as random graphs. The results suggest that systems with many weak interactions are generally more stable, whereas systems with fewer but stronger interactions are more prone to instability unless their structure is carefully managed. These insights provide valuable insights for designing resilient environmental networks and optimizing the management of interconnected natural and engineered systems.
A. Kaveh,
Volume 15, Issue 3 (8-2025)
Abstract
In this paper, a review is provided for the optimal analysis of structures using the graph theoretic force method. An analysis is defined as “optimal” if the corresponding structural matrices (flexibility or stiffness) are sparse, well-structured, and well-conditioned. An expansion process together with the union-intersection theorem is utilized for generating subgraphs, forming a special cycle basis, corresponding to highly localized self equilibration systems. Admissibility checks are used in place of the more common independence checks to speed up the formation of the basis. An efficient solution requires organizing the non-zero entries into various well-defined patterns. Algorithms are provided to form matrices having banded matrices and small profiles. Though the paper considers mainly skeletal structures, the presented concepts are easily extensible to other finite element models. References for such generalizations have been provided. A brief review of swift analysis methods that skirt the harder problem of matrix conditioning is also provided. The iterative nature of optimal structural design via metaheuristic algorithms rewards any speedup in the analysis process. This review recommends utilizing the force method instead of the alternative displacement method to achieve said speedup. The work concludes with a discussion of future challenges in the field of optimal analysis.
