# A Response-Adaptive Method for Design of Validation Experiments in Computational Mechanics
## 1 Introduction  
Computer simulations are widely used to predict the performance of engineered products [1; 2]. It is generally adequate to ensure the accuracy of computer simulation results. To address the need, the concept of verification and validation (V & V) was studied. Several decades ago, the software development community developed V & V guidelines for software quality assurance (SQA). The Institute of Electrical and Electronic Engineers (IEEE) adopted the V & V guidelines for SQA [3]. However, the V & V guidelines were not sufficient to cover the nuances of computational physics and engineering. For example, the V & V guidelines did not consider the details of first principle analyses for computational physics and engineering. To fill the gap, application-specific V & V guidelines were developed. The American Society of Mechanical Engineers (ASME) developed a V & V guideline for validation of computational models in solid mechanics. According to the ASME guideline published in 2006 [4], the goal of validation is \"to determine the predictive capability of a computational model for itsintended use\". As presented in Figure 1, model validation in computational mechanics is defined as a series of activities that aim to maximize the agreement between responses from simulations and experiments [5,6]. In the activity of validity check, it is desirable to prepare a set of simulation and experimental responses that incorporate various uncertainties in product geometry, material properties, and environmental and operational loading. For physical testing in the validation domain, a sufficient number of test samples; ideally, an infinite number of test samples should be prepared. Then, the agreement between the computational and experimental responses can be evaluated without any uncertainty. To this end, a mean-based comparison approach can be used [7]. The mean of experimental responses is compared to the simulation response. If two responses coincide, it is concluded that the computational model is valid. However, in reality, available resources (e.g., time, budget, facilities) for actual testing are always limited. The mean-based comparison approach can provide an incorrect conclusion due to epistemic uncertainty that is attributed to a limited number of test samples. Another approach based on Bayes' theorem was developed to determine the validity of a computational model with a limited number of samples [9,10]. The Bayesian approach was shown to be effective for a particular type of problems, e.g., thermal challenge problem [11]. However, the Bayesian approach is often limited in that initial lack of information due to limited samples should be presented by prior distributions. The prior distributions will deviate depending on the user's domain knowledge and engineering decisions. Therefore, the Bayesian approach cannot be an ideal solution to determine the validity of a computational model with limited samples. Recently, a hypothesis testing-based approach was proposed as an alternative for validity check in model validation [12]. The hypothesis testing-based approach does not require prior distributions. The epistemic uncertainty is represented in the form of type I error in hypothesis testing, which is the probability of incorrectly rejecting a valid computational model (e.g., a false positive). The second type of error, type II error, refers to the probability of improperly accepting an invalid computational model (e.g., a false negative). Type I and II errors can be reduced if additional samples are tested. In safety-related and mission-critical systems, type II errors should be considered more serious than type I errors, as accepting an invalid model can have significant consequences. However, it is not  
Figure 1: Validation activities for computational models in mechanics [8]. feasible to quantify type II errors without knowing the mother distribution of the experiments. Thus, to minimize type II errors, test procedures should be devised in such a way that type II errors are unlikely to occur.