Stochastic modeling of geometrical uncertainties on complex domains, with application to additive manufacturing and brain interface geometries

Highlights

• Complex geometries are challenging to handle in uncertainty quantification.

• We develop a stochastic modeling framework for geometrical uncertainties.

• The approach can accommodate complex geometrical features on the fly.

• We present applications with increasing levels of complexity.

• Brain and 3D printed geometries are considered to test the robustness of the approach.

Abstract

We present a stochastic modeling framework to represent and simulate spatially-dependent geometrical uncertainties on complex geometries. While the consideration of random geometrical perturbations has long been a subject of interest in computational engineering, most studies proposed so far have addressed the case of regular geometries such as cylinders and plates. Here, standard random field representations, such as Karhunen–Loève expansions, can readily be used owing, in particular, to the relative simplicity to construct covariance operators on regular shapes. On the contrary, applying such techniques on arbitrary, non-convex domains remains difficult in general. In this work, we formulate a new representation for spatially-correlated geometrical uncertainties that allows complex domains to be efficiently handled. Building on previous contributions by the authors, the approach relies on the combination of a stochastic partial differential equation approach, introduced to capture salient features of the underlying geometry such as local curvature and singularities on the fly, and an information-theoretic model, aimed to enforce non-Gaussianity. More specifically, we propose a methodology where the interface of interest is immersed into a fictitious domain, and define algorithmic procedures to directly sample random perturbations on the manifold. A simple strategy based on statistical conditioning is also presented to update realizations and prevent self-intersections in the perturbed finite element mesh. We finally provide challenging examples to demonstrate the robustness of the framework, including the case of a gyroid structure produced by additive manufacturing and brain interfaces in patient-specific geometries. In both applications, we discuss suitable parameterization for the filtering operator and quantify the impact of the uncertainties through forward propagation.

Introduction

Random geometrical uncertainties are ubiquitous in many engineering applications. Evidence of so-called random imperfections and their effects on system performance can be found in a broad array of fields, ranging from contact mechanics and tribology [1], to buckling of nanocomposite structures [2], [3], to the design of origami-based metamaterials [4], [5], [6]. Examples can also be found in [7], [8] for parts produced by additive manufacturing, in [9], [10], [11], [12], [13], [14], [15], [16] for the analysis of cylindrical shells under geometrical imperfections, as well as in [2], [3], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27] for composite structures under various types of loading conditions—to list a few. The existence of such imperfections is often attributed to the underlying manufacturing technology, which inevitably introduces deviations from nominal models due to, e.g., precision limitations (see [28], [29] for additive manufacturing techniques), as well as to service conditions. Another potential source of geometrical uncertainties lies into data acquisition chains where raw data sets can be corrupted by post-processing steps. Segmentation in digital image analysis is one relevant example where low contrasts, combined with geometrical complexity, can lead to substantial errors in classification.

In this context, uncertainty quantification techniques have been extensively deployed to investigate the impact of geometrical uncertainties on quantities of interest (such as buckling loads for composite shells, or foldability for origami-type structures; see the aforementioned references), with the aim of better understanding and mitigating system variability. These uncertainties are often modeled as spatially-dependent random parameters and thus, a random field representation must be adopted. A natural way to construct such a representation relies on the definition of a Gaussian model (which is assumed centered without loss of generality) that can eventually be pushed forward to obtain a non-Gaussian model, depending on the retained state space. Regardless of the latter, the consideration of an underlying Gaussian model facilitates the sampling task, as generation techniques for such fields are numerous and well proven (including spectral methods [30], [31], factorization techniques [32], [33], [34], [35], [36], [37], [38], and Karhunen–Loève expansions [39], [40], [41]). In addition, the combination of a Gaussian model and a transport map leads to a low-dimensional parameterization of the resulting non-Gaussian model (that is, the number of hyperparameters in the stochastic model is small) that is more appropriate for inverse identification than more general representations such as polynomial chaos expansions of random fields. From a stochastic modeling standpoint, the methodology requires the construction of a covariance function (for the Gaussian model) describing how the uncertainties are correlated with one another over the domain of interest. For stationary fields indexed on regular geometries such as cylinders, plates, and spheres, covariance functions are usually quite easy to define in closed-form. However, this central task becomes more challenging on complex index sets presenting curved, non-convex features and topological singularities. While standard models can readily be extended to tackle such problems using a parameterization based on a geodesic distance (see, e.g., [42]), their use is practically limited to the case of isotropic correlation structures.

In this work, we formulate an ad hoc random field representation that allows complex domains to be efficiently handled. The methodology builds on the combination of a stochastic partial differential equation approach (SPDE), introduced to capture the salient features of the geometry such as local curvature, non-convexity, and topological singularities, in the definition of the underlying Gaussian random field (which can exhibit an isotropic or an anisotropic correlation structure), and an information-theoretic model, aimed to enforce boundedness on perturbations and hence, non-Gaussianity. This contribution differs from previous works by the authors, which were primarily focused on stochastic constitutive models [43], [44], [45], in that the random field is here defined and directly sampled on a manifold. To address this situation, we present new methodological developments adapted to the modeling of spatially-correlated geometrical uncertainties. More specifically, we first introduce a strategy where the interface of interest is immersed into a fictitious domain in ${\mathbb{R}}^d$. The coefficient in the SPDE filtering operator is then conveniently defined by projecting the gradients of the solutions to (still) fictitious Laplace problems onto the interface. Second, we construct an appropriate algebraic representation and derive a push-forward transport map that ensures well-posedness and provides reasonable modeling flexibility. Third, we propose a simple updating procedure to tackle conflicting perturbations, which result in local self-intersection in the finite element mesh, at a reasonable computational cost using conditional sampling. In order to demonstrate the robustness of the framework, we finally report numerical results pertaining to the modeling and sampling of correlated stochastic perturbations on surfaces with increasing levels of complexity, including the case of a highly porous structure and brain interfaces in patient-specific geometries. In particular, we discuss the parameterization of the stochastic model accounting for both the nominal geometrical model and the process at the origin of the uncertainties.

The rest of this paper is organized as follows. The stochastic model and methodological aspects are first presented in Section 2. We introduce, in particular, the random field representation for geometrical uncertainties, including the stochastic partial differential equation approach, the updating procedure, and the information-theoretic probabilistic model. Numerical applications are then provided and used to analyze the performance of the framework in Section 3. Conclusions are finally drawn in Section 4.