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Efficient uncertainty quantification of a fully nonlinear and dispersive water wave model with random inputs

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Abstract

A major challenge in next-generation industrial applications is to improve numerical analysis by quantifying uncertainties in predictions. In this work we present a formulation of a fully nonlinear and dispersive potential flow water wave model with random inputs for the probabilistic description of the evolution of waves. The model is analyzed using random sampling techniques and nonintrusive methods based on generalized polynomial chaos (PC). These methods allow us to accurately and efficiently estimate the probability distribution of the solution and require only the computation of the solution at different points in the parameter space, allowing for the reuse of existing simulation software. The choice of the applied methods is driven by the number of uncertain input parameters and by the fact that finding the solution of the considered model is computationally intensive. We revisit experimental benchmarks often used for validation of deterministic water wave models. Based on numerical experiments and assumed uncertainties in boundary data, our analysis reveals that some of the known discrepancies from deterministic simulation in comparison with experimental measurements could be partially explained by the variability in the model input. Finally, we present a synthetic experiment studying the variance-based sensitivity of the wave load on an offshore structure to a number of input uncertainties. In the numerical examples presented the PC methods exhibit fast convergence, suggesting that the problem is amenable to analysis using such methods.

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Notes

  1. The gravitational acceleration constant, g, is set at 9.81 m/s\(^2\).

  2. For a treatment of measure theoretic probability theory see, for example, [27].

  3. The package ORTHPOL [41] is used within the Python modules UQToolbox and SpectralToolbox for the construction of polynomials orthogonal with respect to an arbitrary measure.

  4. The kernel density estimation method [58] with a Gaussian kernel was used here to obtain the approximations of the PDFs from samples of the solution.

  5. The theoretical error of the MC method can be expressed by the standard deviation of the MC estimator (18), which is \(\sigma /\sqrt{n}\). This can be approximated inserting the estimated variance \({\bar{\sigma }}\) and then used to test a convergence criterion for the method. Note that the estimate of LHS is often better than the MC estimate. In the example at hand, both mean and variance are time and space dependent, i.e., \(\mu (\mathbf{x},t),\sigma ^2(\mathbf{x},t)\). For the gauge locations \(\{\mathbf{x}_i\}_{i=1}^8\), we defined the convergence criterion by \(\max _i\left( \frac{\Vert \sigma (\mathbf{x}_i,t) \Vert _\infty / \sqrt{n}}{\Vert \mu (\mathbf{x}_i,t) \Vert _\infty }\right) \le 10^{-2}\).

  6. The number of quadrature points for a 1D Gauss rule of polynomial order 5 is 6. The tensorization of this quadrature rule in two dimensions then leads to \(6^2=36\) evaluation points.

  7. The accuracy estimator is defined similarly to the one used in Sect. 5.1.3. For the means and variances \(\mu _i(x,y_c),\sigma ^2_i(x,y=y_c)\) of the harmonic functions \(\{ {\hat{f}}(x,y=y_c) \}_{i=1}^3\), defined along the centerline of the domain \(y=y_c\), the convergence criterion is given by \(\max _i\left( \frac{\Vert \sigma _i(x,y) \Vert _\infty / \sqrt{n}}{\Vert \mu _i(x,y) \Vert _\infty }\right) \le 10^{-2}\).

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Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800, Kgs. Lyngby, Denmark

    Daniele Bigoni & Allan P. Engsig-Karup

  2. Department of Shipping and Marine Technology, Chalmers University of Technology, 412 96, Gothenburg, Sweden

    Claes Eskilsson

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  1. Daniele Bigoni
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  2. Allan P. Engsig-Karup
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  3. Claes Eskilsson
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Correspondence to Daniele Bigoni.

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Bigoni, D., Engsig-Karup, A.P. & Eskilsson, C. Efficient uncertainty quantification of a fully nonlinear and dispersive water wave model with random inputs. J Eng Math 101, 87–113 (2016). https://doi.org/10.1007/s10665-016-9848-8

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