## Adaptive reduction of constitutive model-form error using *a posteriori* error estimation techniques

## Abstract

In engineering practice, models are typically kept as simple as possible for ease of setup and use, computational efficiency, maintenance, and overall reduced complexity to achieve robustness. In solid mechanics, a simple and efficient constitutive model may be favored over one that is more predictive, but is difficult to parameterize, is computationally expensive, or is simply not available within a simulation tool. In order to quantify the modeling error due to the choice of a relatively simple and less predictive constitutive model, we adopt the use of a *posteriori* model-form error-estimation techniques. Based on local error indicators in the energy norm, an algorithm is developed for reducing the modeling error by spatially adapting the material parameters in the simpler constitutive model. The resulting material parameters are not material properties *per se*, but depend on the given boundary-value problem. As a first step to the more general nonlinear case, we focus here on linear elasticity in which the “complex” constitutive model is general anisotropic elasticity and the chosen simpler model is isotropic elasticity. As a result, the algorithm for adaptive error reduction is demonstrated using two examples: (1) A transversely-isotropic plate with hole subjected to tension, and (2) a transversely-isotropic tubemore »

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1455219

- Report Number(s):
- SAND-2018-6265J

Journal ID: ISSN 0045-7825; 664255

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Computer Methods in Applied Mechanics and Engineering

- Additional Journal Information:
- Journal Name: Computer Methods in Applied Mechanics and Engineering; Journal ID: ISSN 0045-7825

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING; a posteriori error estimation; Uncertainty quantification; Model-form error; Model adaptivity

### Citation Formats

```
Bishop, Joseph E., and Brown, Judith Alice. Adaptive reduction of constitutive model-form error using a posteriori error estimation techniques. United States: N. p., 2018.
Web. doi:10.1016/j.cma.2018.06.005.
```

```
Bishop, Joseph E., & Brown, Judith Alice. Adaptive reduction of constitutive model-form error using a posteriori error estimation techniques. United States. doi:10.1016/j.cma.2018.06.005.
```

```
Bishop, Joseph E., and Brown, Judith Alice. Fri .
"Adaptive reduction of constitutive model-form error using a posteriori error estimation techniques". United States. doi:10.1016/j.cma.2018.06.005. https://www.osti.gov/servlets/purl/1455219.
```

```
@article{osti_1455219,
```

title = {Adaptive reduction of constitutive model-form error using a posteriori error estimation techniques},

author = {Bishop, Joseph E. and Brown, Judith Alice},

abstractNote = {In engineering practice, models are typically kept as simple as possible for ease of setup and use, computational efficiency, maintenance, and overall reduced complexity to achieve robustness. In solid mechanics, a simple and efficient constitutive model may be favored over one that is more predictive, but is difficult to parameterize, is computationally expensive, or is simply not available within a simulation tool. In order to quantify the modeling error due to the choice of a relatively simple and less predictive constitutive model, we adopt the use of a posteriori model-form error-estimation techniques. Based on local error indicators in the energy norm, an algorithm is developed for reducing the modeling error by spatially adapting the material parameters in the simpler constitutive model. The resulting material parameters are not material properties per se, but depend on the given boundary-value problem. As a first step to the more general nonlinear case, we focus here on linear elasticity in which the “complex” constitutive model is general anisotropic elasticity and the chosen simpler model is isotropic elasticity. As a result, the algorithm for adaptive error reduction is demonstrated using two examples: (1) A transversely-isotropic plate with hole subjected to tension, and (2) a transversely-isotropic tube with two side holes subjected to torsion.},

doi = {10.1016/j.cma.2018.06.005},

journal = {Computer Methods in Applied Mechanics and Engineering},

number = ,

volume = ,

place = {United States},

year = {2018},

month = {6}

}

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