# On estimating the curvature attributes and strain invariants of deformed surface through radial basis functions

## Abstract

Numerical methods in approximation of derivatives on regular or irregular gridded and scattered data points are of significance in the study of numerous problems in engineering and geoscience. A mathematically accurate and stable method is assessed to compute the differential quantities, while being flexible and easy to implement. Strain-invariant parameters in deformation analysis and curvature attributes in geometrical analysis of approximated surfaces are independent quantities which could be derived numerically through radial basis functions (RBFs) for scattered data points. Approximation of a function or its derivatives through RBFs is highly correlated to its shape parameter which depends on the number and distribution of the particles on the support domain. A procedure to find the optimal shape parameter for RBFs in the influence domain of each nodal point together with the presentation of a manner of calculating the curvature attributes and strain parameters through Gaussian RBFs is proposed here. Results indicate a significant improvement in the approximation accuracy of strain parameters and curvature attributes, while this approximation is more stable when RBFs with optimal shape parameter are implemented rather than the traditional moving least squares.

- Authors:

- K.N. Toosi University of Technology, Department of Geodesy, Faculty of Geodesy and Geomatics Engineering (Iran, Islamic Republic of)

- Publication Date:

- OSTI Identifier:
- 22769361

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 2; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; APPROXIMATIONS; DEFORMATION; ENGINEERING; LEAST SQUARE FIT; STRAINS

### Citation Formats

```
Mehrabi, H., E-mail: hmehrabi@mail.kntu.ac.ir, and Voosoghi, B.
```*On estimating the curvature attributes and strain invariants of deformed surface through radial basis functions*. United States: N. p., 2018.
Web. doi:10.1007/S40314-016-0380-2.

```
Mehrabi, H., E-mail: hmehrabi@mail.kntu.ac.ir, & Voosoghi, B.
```*On estimating the curvature attributes and strain invariants of deformed surface through radial basis functions*. United States. doi:10.1007/S40314-016-0380-2.

```
Mehrabi, H., E-mail: hmehrabi@mail.kntu.ac.ir, and Voosoghi, B. Tue .
"On estimating the curvature attributes and strain invariants of deformed surface through radial basis functions". United States. doi:10.1007/S40314-016-0380-2.
```

```
@article{osti_22769361,
```

title = {On estimating the curvature attributes and strain invariants of deformed surface through radial basis functions},

author = {Mehrabi, H., E-mail: hmehrabi@mail.kntu.ac.ir and Voosoghi, B.},

abstractNote = {Numerical methods in approximation of derivatives on regular or irregular gridded and scattered data points are of significance in the study of numerous problems in engineering and geoscience. A mathematically accurate and stable method is assessed to compute the differential quantities, while being flexible and easy to implement. Strain-invariant parameters in deformation analysis and curvature attributes in geometrical analysis of approximated surfaces are independent quantities which could be derived numerically through radial basis functions (RBFs) for scattered data points. Approximation of a function or its derivatives through RBFs is highly correlated to its shape parameter which depends on the number and distribution of the particles on the support domain. A procedure to find the optimal shape parameter for RBFs in the influence domain of each nodal point together with the presentation of a manner of calculating the curvature attributes and strain parameters through Gaussian RBFs is proposed here. Results indicate a significant improvement in the approximation accuracy of strain parameters and curvature attributes, while this approximation is more stable when RBFs with optimal shape parameter are implemented rather than the traditional moving least squares.},

doi = {10.1007/S40314-016-0380-2},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 2,

volume = 37,

place = {United States},

year = {2018},

month = {5}

}