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A stabilized collocation method based on the efficient gradient reproducing kernel approximations for the boundary value problems
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DOI：10.1016/j.enganabound.2021.08.010.png)
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摘要
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The meshfree strong form direct collocation method (DCM) can address the domain integration issues involved in the Galerkin meshfree methods and outperform them in the efficiency. However, the accuracy and stability of the DCM may not match the Galerkin meshfree methods in solving the complex problems. Moreover, strong form demands the high order derivatives calculation of the approximation function which increases its computational complexity and costs. Therefore, in the present study, we introduce a stabilized collocation method (SCM) associated with the gradient reproducing kernel (GRK) approximations. This method can improve the accuracy and stability of the DCM by proposing the exact numerical integration in the subdomains, and construct the GRK approximations instead of directly taking the reproducing kernel (RK) derivatives to simplify the computational complexity and promote the efficiency. The exact local integration of the SCM is much more efficient than the global integration of the Galerkin meshfree methods which makes the SCM keep the high efficiency as the DCM. Comparison analyses among the RK-CM, GRK-CM, RK-SCM and GRK-SCM are detailedly studied. Numerical simulations demonstrate that GRK-SCM can surpass the conventional RK-CM in accuracy, stability and efficiency, followed by the RK-SCM and then the GRK-CM, which provides an efficient and stable meshfree method for solving the boundary value problems.
Keyword:
Stabilized collocation method
Gradient reproducing kernel approximations
Efficiency
accuracy
Stability
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