Vol.:(0123456789) 1 3 Asian Journal of Civil Engineering https://doi.org/10.1007/s42107-018-00111-z ORIGINAL PAPER Transient Galerkin fnite volume solution of dynamic stress intensity factors Saeed Reza Sabbagh‑Yazdi 1  · Arwin Farhoud 1  · Masoud Zabihi‑Samani 2 Received: 30 October 2018 / Accepted: 31 December 2018 © Springer Nature Switzerland AG 2019 Abstract Transient Galerkin fnite volume method (GFVM) is developed to solve time-dependent problems and analysis of the dynamic stress intensity factors (DSIFs) for cracked problem. An interesting feature of the developed method is its matrix free operations; therefore, it obviously reduces the computation workloads for dynamic cases with small time marching. The two-point displacement extrapolation method is used for calculating the stress intensity factors (SIFs). To show the ability of this method, the structural problem, such as a beam under dynamic load, is considered as the frst case study. The computed transient defections are used for evaluating the accuracy of the GFVM in comparison with the results of the explicit fnite element method (explicit-FEM) and meshless method solvers. A comparison of the CPU time consumption of the GFVM and explicit-FEM solvers shows that the GFVM entails lesser time consumption than the explicit-FEM, without reducing the accuracy of the results. In the second case study, the SIFs are computed for plate with inner crack under constant loading. For the third and fourth case study, the ability of the proposed GFVM solver to cope with DSIFs for a plate with an edge crack and L-shape plate with an inclined crack under dynamic load were tested. The comparison indicates that the GFVM not only provide compatible accuracy close to other common numerical solvers, also ofers considerable CPU-time consumption, in comparison with the methods that requires matrix manipulations. Keywords Galerkin fnite volume method · Dynamic stress intensity factor · Structural elastodynamic problems · Unstructured triangular mesh Introduction The structural analysis problem under dynamic load has always been one of the most important issues in engineer- ing. Considering the importance of this issue, there are only a few dynamic structural problems that are solved with exact and analytical solutions. Therefore, the use of the numerical methods is inevitable. One of the most common and impor- tant methods is the fnite element method (FEM), which is suitable to analyse the elasto-static and elasto-dynamic problems. Furthermore, in fuid–structure interaction (FSI) prob- lems such as the infuence of wind load on structures using the both fnite volume method (FVM) and FEM solvers simultaneously, good achievement has been obtained (Naeiji et al. 2017; Zisis et al. 2017). Due to the problems encoun- tered in the FEM, including volumetric locking during analysis of excessive displacements (Bijelonja et al. 2006) and high-time consumption for time-dependent problems, alternative methods are developing. Although the advent of new numerical methods such as meshless methods (Nayroles et al. 1992; Wang et al. 2002), have provided an appropri- ate answer to computational problems of classical FEM, the computational cost of meshless methods can be higher than the FEM for similar cases. Therefore, time consumption computations of the dynamic problems are still critical tasks for real-world applications, especially for the problems with complex geometry and fne incremental dynamic loading. Previously, the fnite volume method (FVM) was used for solving the computational fuid dynamic (CFD) problems. However, this trend has changed in the recent years and the FVM is used to analyse the computational solid mechanics (CSM) problems as well (Fryer et al. 1991; Suliman et al. * Arwin Farhoud efarhoud@mail.kntu.ac.ir 1 Civil Engineering Department, K.N. Toosi University of Technology, No. 1346, Tehran 19697, Iran 2 Department of Civil Engineering, Parand Branch, Islamic Azad University, Parand, Iran