Fictitious Domain with Least-Squares Spectral Element Method to explore geometric uncertainties by Chaos Collocation

In this paper Chaos Collocation method coupled to Fictitious Domain approach has been applied to oneand two-dimensional elliptic problems defined on random domains in order to demonstrate the accuracy and convergence of the methodology. Chaos Collocation method replaces a stochastic process with a set of deterministic problems, which can be separately solved, so that the big advantage of Chaos Collocation is that it is non-intrusive and existing deterministic solvers can be used. For the analysis of differential problems obtained by Chaos Collocation, Fictitious Domain method with Least-Squares Spectral Element approximation has been employed. This algorithm exploits a fictitious computational domain, where the boundary constraints, immersed in the new simple shaped domain, are enforced by means of Lagrange multipliers. For this reason its main advantage lies in the fact that only a Cartesian mesh, that represents the enclosure, needs to be generated. Excellent accuracy properties of developed method are demonstrated by numerical experiments. 1.0 INTRODUCTION In most engineering applications, to solve physical problems deterministic mathematical models are adopted. It is evident these models are rough simplifications of reality. Actually, many physical input parameters are not deterministic entities, but stochastic processes which certainly influence the behaviour of solution. In order to obtain reliable results uncertainty quantification is necessary and the influence of inherent physical and geometric uncertain parameters must not be neglected. Thanks to the increasing of computer facilities and the advance of algorithms, which leads to more accurate solutions, the current state of technology allows including these inherent uncertainties in mathematical models. There have been an increasing interest in uncertainty analysis applied to computational physics and probabilistic methods have been developed. These methods for uncertainty quantification can be divided into two main categories: non-intrusive, or statistical, and intrusive, or non-statistical. Monte Carlo [3], Stochastic Collocation [16], Chaos Collocation [13] are examples of non-intrusive approaches, Chaos Polynomials [34][36] are examples of intrusive approaches. Non-intrusive methods allow the use of existing deterministic solvers, whereas intrusive approaches need to modify the solver obtaining an efficient tool but limited to solve just a set of problems. In literature there are several examples of numerical methods to face problems with uncertain input parameters and in [14][30][33][34][35] we find applications of these methodologies to thermo-fluid dynamics. In these works the effort is focused on exploring random material properties or random boundary conditions, whereas the topology of domain boundaries are described in deterministic terms, without taking in account of their stochastic nature (Figure 1). Parussini, L. ; Pediroda, V. (2007) Fictitious Domain with Least-Squares Spectral Element Method to Explore Geometric Uncertainties by Chaos Collocation. In Computational Uncertainty in Military Vehicle Design (pp. 36-1 – 36-28). Meeting Proceedings RTO-MP-AVT-147, Paper 36. Neuilly-sur-Seine, France: RTO. Available from: http://www.rto. nato.int. UNCLASSIFIED/UNLIMITED UNCLASSIFIED/UNLIMITED Fictitious Domain with Least-Squares Spectral Element Method to Explore Geometric Uncertainties by Chaos Collocation 36 2 RTO-MP-AVT-147 On the contrary there are really few examples of numerical methods to analyse geometric uncertainty, which means to solve deterministic problems, in terms of material properties and boundary constraints, on random domains (Figure 2), with geometric uncertainty given by shape tolerance. In [11][36], geometric uncertainty pertains to roughness, which is an important thermo-fluid dynamic parameter. In particular [11] presents an analytical model to study the roughness in supersonic flow and the analytical results are compared with several Polynomial Chaos methodologies, such as mapping, Stochastic Collocation and Sparse Grid. In [36] the fluid dynamics in rough channels is studied using a mapping methodology. The obtained results confirm Chaos methods are suitable to study geometric uncertainty given roughness. Figure 1: Differential problem with stochastic material properties and stochastic boundary conditions, where θ is uncertainty. Figure 2: Differential problem with stochastic definition domain, where θ is uncertainty. Another interesting topic pertaining to geometric uncertainty is the study of shape tolerance and its interaction with the state problem. In [8] the influence of geometric tolerance on fluid dynamic field is shown, referring to an angle. A Chaos methodology is used to explore the random domain, in particular a non-intrusive approach. In this work we present a method to face geometric tolerance problems, which allows avoiding the limitations of previous referred methodologies, in particular both the difficulty of mapping space variables into a deterministic domain and the need to remesh the geometry of domain. Pursuing this purpose, we present a new method based on the coupling of Chaos Collocation method and Fictitious Domain approach. According to Generalized Polynomial Chaos theory, a stochastic process can be expressed as a spectral expansion based on suitable orthogonal polynomial with weights associated with a particular density. For non-intrusive approach, a Collocation projection is applied in the random dimension and the resulting system of deterministic equations is then solved to obtain the solution for each random node of nodal approximation. The important advantage of using a non-intrusive method is there is no need to modify the solver of deterministic equations, simplifying the management of computational process. UNCLASSIFIED/UNLIMITED UNCLASSIFIED/UNLIMITED Fictitious Domain with Least-Squares Spectral Element Method to Explore Geometric Uncertainties by Chaos Collocation RTO-MP-AVT-147 36 3 To solve the deterministic problem Least Squares Spectral Element Method is used [4][19][20][21][22] [23]. Moreover a Fictitious Domain approach is adopted, introducing more benefits to methodology. Fictitious Domain methods allow problems formulated on an intricate domain Ω to be solved on a simpler domain Π containing Ω. The extension of original problem to fictitious region Π/ Ω must be chosen so that the solution of extended problem restricted to Ω coincides with the solution of original problem. There are several approaches to implement Fictitious Domain: Elimination method [37], Penalty method [24][37], Distributed Lagrangian method [5][6][7], Boundary Lagrangian method [9][29], Fat Boundary method [17]. A technique which is popular, given its efficiency, is to enforce the boundary conditions by Lagrange multipliers, which is the method we propose in this paper. The main motivation for Fictitious Domain approach is that, defining the extended problem on a simple domain, enables the use of efficient discretization methods on simple structured grids. In this way the solution of state problem is independent by small variations of domain Ω subject to uncertainty and contained into computational domain Π, which is independent by random geometric parameters. Thereby an evident further advantage of the method is that the remeshing of computational domain for each random node solution has no more to be performed. The excellent accuracy of proposed methodology, Chaos Collocation method and Fictitious Domain with Least-Squares Spectral Element approximation, is demonstrated by numerical experiments. The paper is organized as follows. In Section 2.0 some details are given about uncertainty quantification methods. In Section 3.0 the approaches to face problems with geometric uncertainties are discussed. In Section 4.0 the Fictitious Domain method is illustrated and its good accuracy properties are demonstrated by numerical experiments. In Section 5.0 the formulation of Chaos Collocation Method with Fictitious Domain approach is explained and in Section 6.0 some numerical examples to validate the Chaos Collocation Method with Fictitious Domain approach are shown. In Section 7.0 we give some concluding remarks. 2.0 UNCERTAINTY QUANTIFICATION METHODS 2.1 Setting of the problem: stochastic differential equation In this section the Generalized Polynomial Chaos is explained in order to solve the following stochastic differential equation: ( ) ( ) θ φ θ , , ; , , t f t L x x = (1) where L is a differential operator which contains space and time differentiation and can be non linear and depended on random parameters θ , ( ) θ φ , , t x is the solution and function of the space d R ∈ x , time t and random parameters θ ; ( ) θ , ,t f x is a space, time and random parameters dependent source term. 2.2 The Generalized Polynomial Chaos Under specific conditions [28], a stochastic process can be expressed as a spectral expansion based on suitable orthogonal polynomial with weights associated with a particular density. The first study in this field is the Wiener process [31][32], which can be written as a spectral expansion in terms of Hermite polynomials with normal distributed input parameters. The basic idea is to project the variables of the problems on a stochastic space modelled by a set of eigenfunctions Ψ that depend on random variables ) (θ ξ , where θ is the random event. For example, the variable φ has the following spectral finite dimensional representation: UNCLASSIFIED/UNLIMITED UNCLASSIFIED/UNLIMITED Fictitious Domain with Least-Squares Spectral Element Method to Explore Geometric Uncertainties by Chaos Collocation 36 4 RTO-MP-AVT-147 ( ) ( ) ( ) ( ) ∑ ∞ = Ψ = 0 , , , i i i t t θ φ θ φ ξ x x (2) In practical terms the Eq.(2) divides the random variables ( ) θ φ , , t x into a deterministic part, the coefficient ( ) t i , x φ , and a stochastic part, the polynomial chaos ( ) ( ) θ ξ i Ψ . The basis { } i Ψ