where (N_n) is the number of nodes. Quadratic interpolation is essential to resolve the steep gradients of (\phi) within the diffusive crack zone. A goal‑oriented error estimator based on the phase‑field gradient is used:
The manuscript follows the conventional structure (Title, Abstract, Keywords, etc.) and includes all the essential elements (governing equations, numerical algorithm, validation, results, discussion, and references). Feel free to copy the LaTeX source into your favourite editor (Overleaf, TeXShop, etc.) and adapt the figures, tables, or code snippets to your own data. Authors : First Author ¹, Second Author ², Third Author ³ ¹ Department of Mechanical Engineering, University A, City, Country. ² Institute of Applied Mathematics, University B, City, Country. ³ Materials Science Division, Research Center C, City, Country.
where (N_n) is the number of nodes. Quadratic interpolation is essential to resolve the steep gradients of (\phi) within the diffusive crack zone. A goal‑oriented error estimator based on the phase‑field gradient is used:
The manuscript follows the conventional structure (Title, Abstract, Keywords, etc.) and includes all the essential elements (governing equations, numerical algorithm, validation, results, discussion, and references). Feel free to copy the LaTeX source into your favourite editor (Overleaf, TeXShop, etc.) and adapt the figures, tables, or code snippets to your own data. Authors : First Author ¹, Second Author ², Third Author ³ ¹ Department of Mechanical Engineering, University A, City, Country. ² Institute of Applied Mathematics, University B, City, Country. ³ Materials Science Division, Research Center C, City, Country. Working Model 2d Crack-
[ \mathbfu^h(\mathbfx) = \sum_i=1^N_n \mathbfN_i(\mathbfx) , \mathbfu i, \qquad \phi^h(\mathbfx) = \sum i=1^N_n N_i(\mathbfx) , \phi_i, \tag5 ] where (N_n) is the number of nodes