HP-adaptivity - Revision history

E6WikiAdmin at 17:03, 22 January 2024

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Mitjajancic at 07:51, 25 September 2022

2022-09-25T07:51:08Z

Mitjajancic at 07:47, 20 September 2022

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Mitjajancic at 07:06, 20 September 2022

2022-09-20T07:06:11Z

Mitjajancic: Mitjajancic moved page Hp-adaptivity to HP-adaptivity

2022-09-20T05:41:13Z

Mitjajancic moved page Hp-adaptivity to HP-adaptivity

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 {{Box-round|title= Related papers |
-[https://e6.ijs.si/ParallelAndDistributedSystems/publications/153678339.pdf M. Jančič, G. Kosec; Strong form mesh‑free hp‑adaptive solution of linear elasticity problem, Engineering with computers, vol. 39, 2023 DOI: 10.1007/s00366-023-01843-6]
+[https://e6.ijs.si/ParallelAndDistributedSystems/publications/153678339.pdf M. Jančič, G. Kosec; Strong form mesh‑free hp‑adaptive solution of linear elasticity problem, Engineering with computers, vol. 39, 2023 [DOI: 10.1007/s00366-023-01843-6]]
 }}

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 {{Box-round|title= Related papers |
-[https://e6.ijs.si/ParallelAndDistributedSystems/publications/153678339.pdf M. Jančič, G. Kosec; Strong form mesh‑free hp‑adaptive solution of linear elasticity problem, Engineering with computers, vol. 39, 2023 [DOI: 10.1007/s00366-023-01843-6]]
+[https://e6.ijs.si/ParallelAndDistributedSystems/publications/153678339.pdf M. Jančič, G. Kosec; Strong form mesh‑free hp‑adaptive solution of linear elasticity problem, Engineering with computers, vol. 39, 2023 DOI: 10.1007/s00366-023-01843-6]
 }}

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 {{Box-round|title= Related papers |
-[https://e6.ijs.si/ParallelAndDistributedSystems/publications/153678339.pdf M. Jančič, G. Kosec; Strong form mesh‑free hp‑adaptive solution of linear elasticity problem, Engineering with computers, vol. 39, 2023 [DOI: 10.1007/s00366-023-01843-6][COBISS: 153678339]]
+[https://e6.ijs.si/ParallelAndDistributedSystems/publications/153678339.pdf M. Jančič, G. Kosec; Strong form mesh‑free hp‑adaptive solution of linear elasticity problem, Engineering with computers, vol. 39, 2023 [DOI: 10.1007/s00366-023-01843-6]]
 }}

@@ Line 1: / Line 1: @@
 {{Box-round|title= Related papers |
 [https://e6.ijs.si/ParallelAndDistributedSystems/publications/153678339.pdf M. Jančič, G. Kosec; Strong form mesh‑free hp‑adaptive solution of linear elasticity problem, Engineering with computers, vol. 39, 2023 [DOI: 10.1007/s00366-023-01843-6][COBISS: 153678339]]
 }}
 Adaptive solution procedures are essential in problems where the accuracy of the numerical solution varies spatially and are currently subject of intensive studies. Two conceptually different adaptive approaches have been proposed, namely $p$-adaptivity or $h$-, $r$-adaptivity. In $p$-adaptivity, the accuracy of the numerical solution is varied by changing the order of approximation, while in $h$- and $r$-adaptivity the resolution of the spatial discretization is adjusted for the same purpose. In the $h$-adaptive approach, nodes are added or removed from the domain as needed, while in the $r$-adaptive approach, the total number of nodes remains constant - the nodes are only repositioned with respect to the desired accuracy. Ultimately, $h$- and $p$-adaptivities can be combined to form the so-called $hp$-adaptivity, where the accuracy of the solution is controlled with the order of the method and the resolution of the spatial discretization.

← Older revision		Revision as of 17:00, 22 January 2024
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	{{Box-round\|title= Related papers \|		{{Box-round\|title= Related papers \|
	[https://e6.ijs.si/ParallelAndDistributedSystems/publications/153678339.pdf M. Jančič, G. Kosec; Strong form mesh‑free hp‑adaptive solution of linear elasticity problem, Engineering with computers, vol. 39, 2023 [DOI: 10.1007/s00366-023-01843-6][COBISS: 153678339]]		[https://e6.ijs.si/ParallelAndDistributedSystems/publications/153678339.pdf M. Jančič, G. Kosec; Strong form mesh‑free hp‑adaptive solution of linear elasticity problem, Engineering with computers, vol. 39, 2023 [DOI: 10.1007/s00366-023-01843-6][COBISS: 153678339]]

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      \mathcal L u = f_{RHS},
 $$
-where $\mathcal L$ is a differential operator applied to the scalar field $u$ and $f_{RHS}$ is a function. To obtain an error indicator field $\eta$, the problem is first solved implicitly by using a lower order approximation of operators $\mathcal L$, $\mathcal L^{(lo)}_{im}$, obtaining the solution $u^{(im)}$ in the process. The implicitly computed field $u^{(im)}$ is then used to explicitly reconstruct the right-hand side of the problem~\eqref{eq: general PDE}, but this time using a higher order approximation of $\mathcal L$, $\mathcal L^{(hi)}_{ex}$, obtaining $f_{RHS}^{(ex)}$ in the process.
+where $\mathcal L$ is a differential operator applied to the scalar field $u$ and $f_{RHS}$ is a function. To obtain an error indicator field $\eta$, the problem is first solved implicitly by using a lower order approximation of operators $\mathcal L$, $\mathcal L^{(lo)}_{im}$, obtaining the solution $u^{(im)}$ in the process. The implicitly computed field $u^{(im)}$ is then used to explicitly reconstruct the right-hand side of the governing problem, but this time using a higher order approximation of $\mathcal L$, $\mathcal L^{(hi)}_{ex}$, obtaining $f_{RHS}^{(ex)}$ in the process.
 Finally, $f_{RHS}^{(ex)}$ and $f_{RHS}$ are compared $\eta = \left | f_{RHS} - f_{RHS}^{(ex)} \right |$ to indicate the error of the numerical solution.
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 Before the 4 modules can be iteratively repeated, the domain is rediscretized taking into account the newly computed local internodal distances $h_i^{new}(\b p)$ and the local approximation orders $m_i^{new}(\b p)$. However, both are only known in the computational nodes, while global functions $\widehat{h}^{new}(\b p)$ and $\widehat{m}^{new}(\b p)$ over our entire domain space $\Omega$ are required. These are obtained by employing the Sheppard's inverse distance weighting interpolation.
-Figure below schematically demonstrates 3 examples of $hp$-refinements. For the sake of demonstration, the refinement parameters for $h$- and $p$-adaptivity are the same, i.e.\ $\left \{ \alpha, \beta, \lambda, \vartheta \right \} = \left \{ \alpha_h, \beta_h, \lambda_h, \vartheta_h \right \} = \left \{ \alpha_p, \beta_p, \lambda_p, \vartheta_p \right \}$. Additionally, the derefinement aggressiveness $\vartheta$ and the upper threshold $\beta$ are kept constant, so, effectively, only the upper bound of refinement $\alpha$ and refinement aggressiveness $\lambda$ are altered. We observe that the effect refinement parameters is somewhat intuitive. The larger the aggressiveness $\lambda$ the better the local field description and the larger the number of nodes with high approximation order. Similar effect is observed with manipulating the upper refinement threshold $\alpha$, except the effect comes at a smoother manner. Also observe that all refined states were able to increase the accuracy of the numerical solution from the initial state.
+Figure below schematically demonstrates 3 examples of $hp$-refinements. For the sake of demonstration, the refinement parameters for $h$- and $p$-adaptivity are the same, i.e. $\left \{ \alpha, \beta, \lambda, \vartheta \right \} = \left \{ \alpha_h, \beta_h, \lambda_h, \vartheta_h \right \} = \left \{ \alpha_p, \beta_p, \lambda_p, \vartheta_p \right \}$. Additionally, the derefinement aggressiveness $\vartheta$ and the upper threshold $\beta$ are kept constant, so, effectively, only the upper bound of refinement $\alpha$ and refinement aggressiveness $\lambda$ are altered. We observe that the effect refinement parameters is somewhat intuitive. The larger the aggressiveness $\lambda$ the better the local field description and the larger the number of nodes with high approximation order. Similar effect is observed with manipulating the upper refinement threshold $\alpha$, except the effect comes at a smoother manner. Also observe that all refined states were able to increase the accuracy of the numerical solution from the initial state.
 [[File:refinement_demonstration.png|800px]]
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 [[File:fwo_solution_hp.png|800px]]
-For a detailed analysis, we look at the surface traction $\sigma_{xx}$, as it is often used to determine the location of crack initiation. The surface traction is shown in Figure~\ref{fig:fwo_conv} for $6$ selected adaptivity iterations. The mesh-free nodes are colored according to the local approximation order enforced by the $hp$-adaptive solution procedure. The message of this figure is twofold. Firstly, it is clear that the proposed IMEX error indicator can be successfully used in linear elasticity problems, and secondly, we observe that the $hp$-adaptive solution procedure successfully approximated the surface traction in the neighborhood of the contact. In the process, the local field description under the contact has been significantly improved and also the local approximation orders have taken a non-trivial distribution.
+For a detailed analysis, we look at the surface traction $\sigma_{xx}$, as it is often used to determine the location of crack initiation. The surface traction is shown in figure below for $6$ selected adaptivity iterations. The mesh-free nodes are colored according to the local approximation order enforced by the $hp$-adaptive solution procedure. The message of this figure is twofold. Firstly, it is clear that the proposed IMEX error indicator can be successfully used in linear elasticity problems, and secondly, we observe that the $hp$-adaptive solution procedure successfully approximated the surface traction in the neighborhood of the contact. In the process, the local field description under the contact has been significantly improved and also the local approximation orders have taken a non-trivial distribution.
 [[File:fwo_abaqus.png|800px]]
-The surface traction in Figure~\ref{fig:fwo_conv} is additionally accompanied with the FEM results on a much denser mesh with more than 100\,000 DOFs obtained with the commercial solver Abaqus\textsuperscript{\textregistered}. To compute the absolute difference between the two methods, the mesh-free solution has been interpolated to Abaqus's computational points using the Sheppard's inverse distance weighting interpolation with $2$ closest neighbors. We see that the absolute difference is decreasing with the number of adaptivity iterations, finally settling at approximately 2 \% of the maximum value under the contact at initial iteration. The highest absolute difference is expectedly located at the edges of the contact, i.e.\ around $x=a$ and $x=-a$, while the difference in the rest of the contact is even smaller.
+The surface traction is additionally accompanied with the FEM results on a much denser mesh with more than 100\,000 DOFs obtained with the commercial solver Abaqus\textsuperscript{\textregistered}. To compute the absolute difference between the two methods, the mesh-free solution has been interpolated to Abaqus's computational points using the Sheppard's inverse distance weighting interpolation with $2$ closest neighbors. We see that the absolute difference is decreasing with the number of adaptivity iterations, finally settling at approximately 2 % of the maximum value under the contact at initial iteration. The highest absolute difference is expectedly located at the edges of the contact, i.e. around $x=a$ and $x=-a$, while the difference in the rest of the contact is even smaller.

← Older revision		Revision as of 07:06, 20 September 2022
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← Older revision	Revision as of 05:41, 20 September 2022
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