In this paper we address the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo methods for large-scale applications with high dimensional parameter spaces, e.g. in uncertainty quantification in porous media flow. We propose a new multilevel Metropolis-Hastings algorithm, and give an abstract, problem dependent theorem on the cost of the new multilevel estimator based on a set of simple, verifiable assumptions. For a typical model problem in subsurface flow, we then provide a detailed analysis of these assumptions and show significant gains over the standard Metropolis-Hastings estimator. Numerical experiments confirm the analysis and demonstrate the effectiveness of the method with consistent reductions of more than an order of magnitude in the cost of the multilevel estimator over the standard Metropolis-Hastings algorithm for tolerances ε < 10 −2 . for any q < ∞ and δ > 0, where the (generic) constant C k,f,ψ,q (here and below) depends on the data k, f , ψ and on q, but is independent of any other parameters.Proof. This follows from [34, Proposition 4.1].Convergence results for functionals of the solution p can now be derived from Theorem 4.2 using a duality argument. We will here for simplicity only consider bounded, linear functionals, but the results extend to continuously Frèchet differentiable functionals (see [34, §3.2]). We make the following assumption on the functional G (cf. Assumption F1 in [34]).A2. Let G : H 1 (D) → R be linear, and suppose there exists C G ∈ R, such thatfor all δ > 0.An example of a functional which satisfies A2 is a local average of the pressure, 1 |D * | D * p dx for some D * ⊂ D. The main result on the convergence for functionals is the following. Corollary 4.3. Let the assumptions of Theorem 4.2 be satisfied, and suppose G satisfies A2. Thenfor any q < ∞ and δ > 0.Proof. This follows from [34, Corollary 4.1].Note that assumption A2 is crucial in order to get the faster convergence rates of the spatial discretisation error in Corollary 4.3. For multilevel estimators based on i.i.d. samples, it follows immediately from Corollary 4.3 that the (corresponding) assumptions M1 and M2 are satisfied, with α = 1/d + δ, α = 1/2 + δ and β = 2α, β = 2α , for any δ > 0 (see [34] for details).
a b s t r a c tIf carbon fibre layers are prevented from slipping over one another as they consolidate onto a non-trivial geometry, they can be particularly susceptible to wrinkling/buckling instabilities. A one dimensional model for out-of-plane ply wrinkling during consolidation over an external radius is presented. Critical conditions for the appearance of wrinkles provide manufacturing strategies to eliminate such defects. Predicted wrinkle wavelengths and critical wrinkling conditions show good agreement with wrinkle defects observed in a spar demonstrator.
In this paper we derive an obstacle problem with a free boundary to describe the formation of voids at areas of intense geological folding. An elastic layer is forced by overburden pressure against a V-shaped rigid obstacle. Energy minimization leads to representation as a nonlinear fourth-order ordinary differential equation, for which we prove their exists a unique solution. Drawing parallels with the Kuhn-Tucker theory, virtual work, and ideas of duality, we highlight the physical significance of this differential equation. Finally we show this equation scales to a single parametric group, revealing a scaling law connecting the size of the void with the pressure/stiffness ratio. This paper is seen as the first step towards a full multilayered model with the possibility of voiding.
In this paper we address the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo methods for large-scale applications with high-dimensional parameter spaces, e.g., in uncertainty quantification in porous media flow. We propose a new multilevel Metropolis-Hastings algorithm and give an abstract, problem-dependent theorem on the cost of the new multilevel estimator based on a set of simple, verifiable assumptions. For a typical model problem in subsurface flow, we then provide a detailed analysis of these assumptions and show significant gains over the standard Metropolis-Hastings estimator. Numerical experiments confirm the analysis and demonstrate the effectiveness of the method with consistent reductions of more than an order of magnitude in the cost of the multilevel estimator over the standard Metropolis-Hastings algorithm for tolerances ε < 10 −2 .
Understanding the inter-ply shear behaviour of uncured carbon fibre prepreg is fundamental to avoiding process-induced defects during manufacturing of largescale components. Shear tests for AS4/8552 are compared to a one-dimensional viscoelastic-plastic model for inter-ply shear. The paper presents a methodology capable of determining the parameters of temperature, rate and pressure required for minimum resistance to movement of a prepreg. Investigating the joint strength and friction values individually shows that friction increases with temperature, contrary to previous work, and that the new value of joint strength is predominant at lower temperatures. Rate dependent variables are strongly linked to the resin behaviour, confirming the need for a viscoelastic model. Simple application to industrial scenarios is discussed along with more complex process modelling.
Large aerospace parts are typically certified by testing narrow specimens, such as curved laminates, which have exposed free edges. These edges (not present in the production part) have been found to reduce the 3D strength of curved laminates by over 20%, showing this certification method is unreasonably conservative. The free edges also create a singularity, such that Finite Element (FE) modelling is challenging, which is typically approximated using non-linear analysis of cohesive interlaminar zones. A new treatment process is developed whereby a layer of resin is applied to the free edges of curved laminates. This significantly reduces the edge effect and delays failure. The resin edge treatment increases the strength of the curved laminate test specimens by 16%. The treatment also simplifies FE modelling by allowing for non-zero stresses normal to the laminate edge, removing the singularity. This enables use of linear FE models, which converge at the laminate edge. A linear FE method developed in this paper is conservative and predicts the strength of treated curved laminates to within 5% of the average test value. Hence it is shown that the resin edge treatment can be used to improve reliability of both certification tests and FE models.
In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a partition of unity are presented. These new spaces have advantages over those proposed in [I. Babuska and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373-406]. First, in addition to a nearly exponential decay rate of the local approximation errors with respect to the dimensions of the local spaces, the rate of convergence with respect to the size of the oversampling region is also established. Second, the theoretical results hold for problems with mixed boundary conditions defined on general Lipschitz domains. Finally, an efficient and easy-to-implement technique for generating the discrete A-harmonic spaces is proposed which relies on solving an eigenvalue problem associated with the Dirichlet-to-Neumann operator, leading to a substantial reduction in computational cost. Numerical experiments are presented to support the theoretical analysis and to confirm the effectiveness of the new method.
Finite element (FE) analysis has the potential to offset much of the expensive experimental testing currently required to certify aerospace laminates. However, large numbers of degrees of freedom are necessary to model entire aircraft components whilst accurately resolving micro-scale defects. The new module dune-composites, implemented within DUNE by the authors, provides a tool to efficiently solve large-scale problems using novel iterative solvers. The key innovation is a preconditioner that guarantees a constant number of iterations regardless of the problem size. Its robustness has been shown rigorously in Spillane et al. (Numer. Math. 126, 2014) for isotropic problems. For anisotropic problems in composites it is verified numerically for the first time in this paper. The parallel implementation in DUNE scales almost optimally over thousands of cores. To demonstrate this, we present an original numerical study, varying the shape of a localised wrinkle and the effect this has on the strength of a curved laminate. This requires a high-fidelity mesh containing at least four layers of quadratic elements across each ply and interface layer, underlining the need for dunecomposites, which can achieve run times of just over 2 minutes on 2048 cores for realistic composites problems with 173 million degrees of freedom.
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