{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T03:22:09Z","timestamp":1773199329993,"version":"3.50.1"},"reference-count":55,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2023,9,29]],"date-time":"2023-09-29T00:00:00Z","timestamp":1695945600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Education","award":["ISP23-65"],"award-info":[{"award-number":["ISP23-65"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>This paper introduces a robust algorithm that efficiently generates high-quality unstructured triangular meshes to model complex two-dimensional crack growth problems within the framework of linear elastic fracture mechanics (LEFM). The proposed Visual Fortran code aims to address key challenges in mesh generation including geometric complexity, required simulation accuracy, and computational resource constraints. The algorithm incorporates adaptive refinement and updates to the mesh structure near the crack tip, resulting in the formation of rosette elements that provide accurate approximations of stress intensity factors (SIFs). By utilizing the maximum circumferential stress theory, the algorithm predicts the new crack path based on these SIFs. Throughout the simulation of crack propagation, a node splitting approach was employed to represent the progression of the crack, while the crack growth path is determined by successive linear extensions for each crack growth increment. To compute stress intensity factors (SIFs) for each increment of crack extension, a displacement extrapolation method was used. The experimental and numerical results demonstrated the algorithm\u2019s effectiveness in accurately predicting crack growth and facilitating reliable stress analysis for complex crack growth problems in two dimensions. The obtained results for the SIF were found to be consistent with other analytical solutions for standard geometries.<\/jats:p>","DOI":"10.3390\/ma16196481","type":"journal-article","created":{"date-parts":[[2023,9,29]],"date-time":"2023-09-29T07:42:08Z","timestamp":1695973328000},"page":"6481","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Robust Adaptive Mesh Generation Algorithm: A Solution for Simulating 2D Crack Growth Problems"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1599-8833","authenticated-orcid":false,"given":"Abdulnaser","family":"Alshoaibi","sequence":"first","affiliation":[{"name":"Mechanical Engineering Department, College of Engineering, Jazan University, KSA, 114 Almarefah Rd., Jizan 45142, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4510-456X","authenticated-orcid":false,"given":"Yahya","family":"Fageehi","sequence":"additional","affiliation":[{"name":"Mechanical Engineering Department, College of Engineering, Jazan University, KSA, 114 Almarefah Rd., Jizan 45142, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"106918","DOI":"10.1016\/j.engfracmech.2020.106918","article-title":"Numerical modeling of crack propagation with dynamic insertion of cohesive elements","volume":"227","author":"Bouchard","year":"2020","journal-title":"Eng. Fract. Mech."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/j.tafmec.2017.09.012","article-title":"Crack propagation modelling using the weak form quadrature element method with minimal remeshing","volume":"93","author":"Liao","year":"2018","journal-title":"Theor. Appl. Fract. Mech."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1115\/1.3625816","article-title":"Plane problems of cracks in dissimilar media","volume":"32","author":"Rice","year":"1965","journal-title":"J. Appl. Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"45","DOI":"10.30880\/ijie.2019.11.06.006","article-title":"Comprehensive comparisons of two and three dimensional numerical estimation of stress intensity factors and crack propagation in linear elastic analysis","volume":"11","author":"Alshoaibi","year":"2019","journal-title":"Int. J. Integr. Eng."},{"key":"ref_5","first-page":"97","article-title":"Fatigue life and crack path prediction in 2D structural components using an adaptive finite element strategy","volume":"3","author":"Alshoaibi","year":"2008","journal-title":"Int. J. Mech. Mater. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Alshoaibi, A.M., and Fageehi, Y.A. (2021). Adaptive Finite Element Model for Simulating Crack Growth in the Presence of Holes. Materials, 14.","DOI":"10.3390\/ma14185224"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.engfracmech.2013.06.006","article-title":"Element-wise fracture algorithm based on rotation of edges","volume":"110","author":"Areias","year":"2013","journal-title":"Eng. Fract. Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1002\/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J","article-title":"A finite element method for crack growth without remeshing","volume":"46","author":"Dolbow","year":"1999","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"6183","DOI":"10.1016\/S0045-7825(01)00215-8","article-title":"Modeling holes and inclusions by level sets in the extended finite-element method","volume":"190","author":"Sukumar","year":"2001","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_10","first-page":"153","article-title":"Finite elements in fracture mechanics","volume":"201","author":"Kuna","year":"2013","journal-title":"Solid Mech. Its Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.ijsolstr.2018.04.014","article-title":"Local stress evaluation of rapid crack propagation in finite element analyses","volume":"144","author":"Yanagimoto","year":"2018","journal-title":"Int. J. Solids Struct."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/0168-874X(89)90008-5","article-title":"An interactive approach to local remeshing around a propagating crack","volume":"5","author":"Wawrzynek","year":"1989","journal-title":"Finite Elem. Anal. Des."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1016\/j.cma.2017.10.012","article-title":"A 3D h-adaptive local remeshing technique for simulating the initiation and propagation of cracks in ductile materials","volume":"330","author":"Yang","year":"2018","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/j.ijfatigue.2012.11.003","article-title":"A new enriched finite element for fatigue crack growth","volume":"48","author":"Hu","year":"2013","journal-title":"Int. J. Fatigue"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"109017","DOI":"10.1016\/j.engfracmech.2022.109017","article-title":"A coupled XFEM fatigue modelling of crack growth, delamination and bridging in FRP strengthened metallic plates","volume":"279","author":"Rashnooie","year":"2023","journal-title":"Eng. Fract. Mech."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"103890","DOI":"10.1016\/j.finel.2022.103890","article-title":"A singular crack tip element based on sub-partition and XFEM for modeling crack growth in plates and shells","volume":"215","author":"Xing","year":"2023","journal-title":"Finite Elem. Anal. Des."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"107558","DOI":"10.1016\/j.ijfatigue.2023.107558","article-title":"A phase field model for high-cycle fatigue: Total-life analysis","volume":"170","author":"Golahmar","year":"2023","journal-title":"Int. J. Fatigue"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"105118","DOI":"10.1016\/j.jmps.2022.105118","article-title":"Phase-field fracture modeling for large structures","volume":"171","author":"Lo","year":"2023","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_19","unstructured":"Taylor, R., and Zienkiewicz, O. (2013). The Finite Element Method, Butterworth-Heinemann."},{"key":"ref_20","unstructured":"Zienkiewicz, O.C., Taylor, R.L., and Zhu, J.Z. (2005). The Finite Element Method: Its Basis and Fundamentals, Elsevier."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.compstruc.2013.04.004","article-title":"Dynamic grid for mesh generation by the advancing front method","volume":"123","author":"Lo","year":"2013","journal-title":"Comput. Struct."},{"key":"ref_22","unstructured":"Cheng, S.-W., Dey, T.K., and Shewchuk, J. (2012). Delaunay Mesh Generation, CRC Press."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"108191","DOI":"10.1016\/j.ijmecsci.2023.108191","article-title":"An adaptive finite element method for crack propagation based on a multifunctional super singular element","volume":"247","author":"Wang","year":"2023","journal-title":"Int. J. Mech. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/j.engfracmech.2015.10.042","article-title":"Damage and fracture algorithm using the screened Poisson equation and local remeshing","volume":"158","author":"Areias","year":"2016","journal-title":"Eng. Fract. Mech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1016\/j.engfracmech.2017.11.017","article-title":"Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation","volume":"189","author":"Areias","year":"2018","journal-title":"Eng. Fract. Mech."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/j.cma.2012.07.017","article-title":"An adaptive singular ES-FEM for mechanics problems with singular field of arbitrary order","volume":"253","author":"Liu","year":"2013","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/0013-7944(95)00247-2","article-title":"Quasi-automatic simulation of crack propagation for 2D LEFM problems","volume":"55","author":"Bittencourt","year":"1996","journal-title":"Eng. Fract. Mech."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1016\/S0045-7825(99)00324-2","article-title":"Crack propagation modelling using an advanced remeshing technique","volume":"189","author":"Bouchard","year":"2000","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/nme.3163","article-title":"Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture","volume":"92","author":"Park","year":"2012","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2921","DOI":"10.1016\/j.engfracmech.2008.01.006","article-title":"Modeling of crack propagation via an automatic adaptive mesh refinement based on modified superconvergent patch recovery technique","volume":"75","author":"Khoei","year":"2008","journal-title":"Eng. Fract. Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.engfracmech.2013.11.005","article-title":"3-D simulation of arbitrary crack growth using an energy-based formulation\u2014Part I: Planar growth","volume":"115","author":"Davis","year":"2014","journal-title":"Eng. Fract. Mech."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/S0168-874X(99)00057-8","article-title":"A simple finite element formulation for computing stress singularities at bimaterial interfaces","volume":"35","author":"Sze","year":"2000","journal-title":"Finite Elem. Anal. Des."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Alshoaibi, A.M., and Fageehi, Y.A. (2022). A Computational Framework for 2D Crack Growth Based on the Adaptive Finite Element Method. Appl. Sci., 13.","DOI":"10.3390\/app13010284"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1115\/1.4011547","article-title":"Analysis of stresses and strains near the end of a crack transversing a plate","volume":"24","author":"Irwin","year":"1957","journal-title":"J. Appl. Mech."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"A49","DOI":"10.1115\/1.4008919","article-title":"Bearing pressures and cracks: Bearing pressures through a slightly waved surface or through a nearly flat part of a cylinder, and related problems of cracks","volume":"6","author":"Westergaard","year":"1939","journal-title":"J. Appl. Mech."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Perez, N. (2017). Fracture Mechanics, Springer.","DOI":"10.1007\/978-3-319-24999-5"},{"key":"ref_37","unstructured":"Owen, D.R.J., and Fawkes, A. (1983). Engineering Fracture Mechanics: Numerical Methods and Applications, Pineridge Press Ltd."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1002\/nme.1620030407","article-title":"An automatic mesh generation scheme for plane and curved surfaces by \u2018isoparametric\u2019 co-ordinates","volume":"3","author":"Zienkiewicz","year":"1971","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1002\/nme.1620080402","article-title":"Automatic triangulation of arbitrary planar domains for the finite element method","volume":"8","author":"Cavendish","year":"1974","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_40","first-page":"537","article-title":"Finite element modeling of fatigue crack propagation using a self adaptive mesh strategy","volume":"2","author":"Alshoaibi","year":"2008","journal-title":"Int. Rev. Mech. Eng. (IREME)"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"7850","DOI":"10.1016\/j.jmrt.2020.04.083","article-title":"2D finite element simulation of mixed mode fatigue crack propagation for CTS specimen","volume":"9","author":"Alshoaibi","year":"2020","journal-title":"J. Mater. Res. Technol."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Alshoaibi, A.M., and Fageehi, Y.A. (2021). Simulation of Quasi-Static Crack Propagation by Adaptive Finite Element Method. Metals, 11.","DOI":"10.3390\/met11010098"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1631\/jzus.2007.A0228","article-title":"An adaptive finite element procedure for crack propagation analysis","volume":"8","author":"Alshoaibi","year":"2007","journal-title":"J. Zhejiang Univ.-Sci. A"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"283","DOI":"10.12989\/sem.2010.35.3.283","article-title":"Finite element procedures for the numerical simulation of fatigue crack propagation under mixed mode loading","volume":"35","author":"Alshoaibi","year":"2010","journal-title":"Struct. Eng. Mech."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1007\/BF00037924","article-title":"The natural isoparametric triangle versus collapsed quadrilateral for elastic crack analysis","volume":"12","author":"Freese","year":"1976","journal-title":"Int. J. Fract."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Alshoaibi, A.M., and Bashiri, A.H. (2022). Adaptive finite element modeling of linear elastic fatigue crack growth. Materials, 15.","DOI":"10.3390\/ma15217632"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Bashiri, A.H., and Alshoaibi, A.M. (2020). Adaptive Finite Element Prediction of Fatigue Life and Crack Path in 2D Structural Components. Metals, 10.","DOI":"10.3390\/met10101316"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Anderson, T.L. (2017). Fracture Mechanics: Fundamentals and Applications, CRC Press.","DOI":"10.1201\/9781315370293"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1016\/0013-7944(95)00214-6","article-title":"Strain gage methods for measurement of opening mode stress intensity factor","volume":"55","author":"Parnas","year":"1996","journal-title":"Eng. Fract. Mech."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1016\/j.nucengdes.2004.10.005","article-title":"Experimental investigation on ductile stable crack growth emanating from wire-cut notch in AISI 4340 steel","volume":"235","author":"Mourad","year":"2005","journal-title":"Nucl. Eng. Des."},{"key":"ref_51","unstructured":"Broek, D. (2012). Elementary Engineering Fracture Mechanics, Springer Science & Business Media."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/S0167-8442(97)00038-4","article-title":"A two-strain-gage technique for determining mode I stress-intensity factor","volume":"28","author":"Wei","year":"1997","journal-title":"Theor. Appl. Fract. Mech."},{"key":"ref_53","unstructured":"Ingraffea, A.R., and Grigoriu, M. (1990). Probabilistic Fracture Mechanics: A Validation of Predictive Capability, Cornell University, Department of Structural Engineering."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"115015","DOI":"10.1016\/j.cma.2022.115015","article-title":"An automated mesh generation algorithm for simulating complex crack growth problems","volume":"398","author":"Mohmadsalehi","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"3887","DOI":"10.1016\/S0045-7825(03)00391-8","article-title":"Numerical modelling of crack propagation: Automatic remeshing and comparison of different criteria","volume":"192","author":"Bouchard","year":"2003","journal-title":"Comput. Methods Appl. Mech. Eng."}],"container-title":["Materials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1996-1944\/16\/19\/6481\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:01:52Z","timestamp":1760130112000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1996-1944\/16\/19\/6481"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,29]]},"references-count":55,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["ma16196481"],"URL":"https:\/\/doi.org\/10.3390\/ma16196481","relation":{},"ISSN":["1996-1944"],"issn-type":[{"value":"1996-1944","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,29]]}}}