An h-p adaptive method using clouds

TLDR: It is shown how h, p and h- p adaptivity can be implemented in the h-p cloud method without traditional grid concepts typical of finite element methods.

About: This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 1996-12-15 and is currently open access. It has received 917 citations till now. The article focuses on the topics: Finite element method & Cloud computing.

Figures

Fig. 19. Pointwise error in the radial component of the displacement vector for the h-p cloud solution corresponding to p = 4.

Fig. 10. It can be observed in Fig. 10 that the clouds with higher-order approximation are those in the part of the domain where the solution has a strong gradient.

Fig. 5 0.9745 0.9923 0.9930 0.8631 0.8509 34 Fig. 12 0.00765 0.04133 0.04071 0.00282 0.02756 527 Fig. 9 0.6924 0.8529 1.1299 0.6898 0.7669 89 Fig. 10 0.00234 0.01878 0.01784 0.00069 0.01009 944

Fig. 11 shows the three-dimensional plot of the solution obtained using the discretization shown in

Frequently asked questions

Q1. What are the contributions in "An h-p adaptive method using clouds" ?

Several computational and mathematical features of the h-p cloud method are demonstrated in this paper. The authors show how h, p and h-p adaptivity can be implemented in the h-p cloud method without traditional grid concepts typical of finite element methods. 

Q2. What is the important situation where the judicious choice of the weighting functions can?

One important situation where the judicious choice of the weighting functions can be beneficial is in the imposition of essential boundary conditions. 

Q3. What is the problem that needs further study?

One issue needing further study is the development of precise and computationally efficient numerical integration schemes of the h-p cloud functions. 

Q4. What are the different types of methods used for computer simulation of complex problems?

They have been categorized under diverse headings such as meshless methods, particle methods, wavelet-type methods, elementfree methods, finite point methods, etc. 

Q5. What is the set of differential equations governing the solutions of this problem?

The set of differential equations governing the solutions of this problem are those from threedimensional elasticity specialized for the axisymmetric case [5,21]. 

Q6. What is the fundamental idea in the h-p cloud method?

The fundamental idea in the h-p cloud method is the construction of the family of functions S:’ using the partition of unity YN defined in the previous section. 

Q7. What are the choices for the h-p cloud method?

Mathematical and numerical analysis performed by Duarte and Oden [lO,ll] have shown that the family of functions 9k=“.p are the best choice for the h-p cloud method. 

Q8. What are the properties of the h-p cloud method?

The resulting functions, called h-p clouds, retain good properties of the MLSF, such as high regularity and compactness, and linear combinations of these functions can represent polynomials of any degree. 

Q9. What is the boundary condition for the h-p cloud?

If the authors set up, = u(x,) VxP E QN, then(2.15)Now suppose that the authors want to impose the following boundary conditionUhP = u - on& (2.16)If (2.15) is true for at least one node xP E r,, the boundary condition (2.16) can be imposed following the approach used in the p and h-p version of the finite element method or in the spectral method. 

Q10. What is the property of h-p clouds?

This property allows the implementation of p and h-p adaptivity in the h-p clouds context with the same remarkable features of h-p finite element methods but without the burden of a mesh. 

Q11. What is the implementation of p adaptivity in the h-p cloud method?

As in the finite element method, the implementation of p adaptivity in the h-p cloud method is easier than the implementation of h or h-p adaptivity. 

Q12. How are the h-p cloud functions set?

In all problems analyzed, the size of the supports of the h-p cloud functions are set by first imposing the condition that every quadrature point belongs to the support of at least one cloud. 

Q13. What is the method for calculating the h-p cloud?

Methods Appl. Mech. Engrg. 139 (1996) 237-262hp cloud: k=O, p=O-4. p- FE: p= I-4 I1 - ‘% .-._. _._.*._._.n hp Clouds -6. t p=o . . .._ ‘Si p version FEM -f3-nu = 0.4999a IIO 100 NFig. 18. 

Q14. What is the h adaptivity of the h-p cloud method?

The following algorithm implements h adaptivity in the h-p cloud method (two-dimensional version):>=765432I0Fig. .5. Discretization used in the step 0 of h, p and h-p adaptation.