Timothy Dodwell

A Hierarchical Multilevel Markov Chain Monte Carlo Algorithm with Applications to Uncertainty Quantification in Subsurface Flow

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

Multi-layered folding with voids.

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SELF-SIMILAR VOIDING SOLUTIONS OF A SINGLE LAYERED MODEL OF FOLDING ROCKS ∗

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On the nucleation and growth of kink and shear bands

In the deformation of layered materials such as geological strata, or stacks of paper, mechanical properties compete with the geometry of layering. Smooth, rounded corners lead to voids between the layers, while close packing of the layers results in geometrically induced curvature singularities. When voids are penalized by external pressure, the system is forced to trade off these competing effects, leading to sometimes striking periodic patterns. In this paper, we construct a simple model of geometrically nonlinear multi-layered structures under axial loading and pressure confinement, with non-interpenetration conditions separating the layers. Energy minimizers are characterized as solutions of a set of fourth-order nonlinear differential equations with contact-force Lagrange multipliers, or equivalently of a fourth-order free-boundary problem. We numerically investigate the solutions of this free-boundary problem and compare them with the periodic solutions observed experimentally.

Internal wrinkling instabilities in layered media

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 there 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 that 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 toward a full multilayered model with the possibility of voiding.

Anti-symmetric laminates for improved consolidation and reduced warp of tapered C-sections

Similarities and differences between the phenomena of kink banding in compressed layered structures and shear banding in compressed granular media are explored. Simple models are introduced for both, and the focus is directed onto how they can nucleate from the perfectly flat state. A convincing scenario is found for each in which a mode develops from an initial bifurcation into a periodic state, followed by rapid localization under falling load, while retaining decaying but wavy tails. At a certain lower critical load, the tails lose their waviness, and the expected form of the kink or shear band appears. In each case, good numerical evidence is provided for the existence of this form of behaviour. A second potential instability for the layered case is also explored, linked to the appearance of a critical force dipole that overcomes bending stiffness locally at some point along the length. This mode, which should appear with non-wavy decaying tails at the lower of the two critical loads mentioned earlier, proves somewhat elusive. Evidence is found for its existence in the linearized approximation to the layered model, but the search for numerical solutions to the underlying nonlinear equation is hindered by a shortage of suitable boundary conditions.

Convoluted accommodation structures in folded rocks

The large deformation Cosserat continuum model presented shows the potential of capturing internal buckling instabilities within layered media at a fraction of the computational cost when compared with conventional modelling methods. A homogenized elasto-plastic Cosserat tensor is derived for periodic arrangements of orthotropic elastic layers separated by weak interfaces that exhibit a modified Mohr–Coulomb friction law. Focus is given to the physical interpretation of the Cosserat deformation measures, the derivation of the Cosserat elastic tensor and implementation of the model using the finite element method. The resulting formulation is validated for three simplified loading scenarios: a cantilever, internal buckling and simple shear of a multilayered beam. Finally, the model is applied to a new application to capture the formation of wrinkling defects during the manufacturing of composite laminates. The results show good agreement with observed manufacturing defects, demonstrating the clear potential for application of the Cosserat model within composite process modelling and other layered material applications.

Uncertainty quantification of composite structures with defects using multilevel monte carlo simulations

Abstract This paper challenges the conventional use of symmetric stacking sequences by proposing fully uncoupled anti-symmetric laminates that simultaneously improve consolidation and reduce warp of tapered laminates. A demonstrator C-section laminate with an anti-symmetric stacking sequence has been designed to eliminate coupling between in-plane and out-of-plane deformation. Results are compared with a conventional, symmetric baseline. The two laminate designs were deposited on a male tool using automated fiber placement and final de-bulk and cure were performed in a single autoclave operation. It is postulated that long continuous plies spanning a tapered region can inhibit consolidation in unidirectional material when they are not equidistant from the tool surface. The ply termination strategy of the anti-symmetric laminate mitigates this effect. It has been found that the anti-symmetric demonstrator achieved better consolidation compared with the baseline symmetric demonstrator. The results also show for the first time, that the level of warping exhibited by the anti-symmetric laminate is approximately 50% lower than the symmetric baseline.

dune-composites – A New Framework for High-Performance Finite Element Modelling of Laminates

A simplified variational model for the formation of convoluted accommodation structures, as seen in the hinge zones of larger-scale geological folds, is presented. The model encapsulates some important and intriguing nonlinear features, notably: infinite critical loads, formation of plastic hinges, and buckling on different length-scales. An inextensible elastic beam is forced by uniform overburden pressure and axial load into a V-shaped geometry dictated by formation of a plastic hinge. Using variational methods developed by Dodwell et al., upon which this paper leans heavily, energy minimisation leads to representation as a fourth-order nonlinear differential equation with free boundary conditions. Equilibrium solutions are found using numerical shooting techniques. Under the Maxwell stability criterion, it is recognised that global energy minimisers can exist with convoluted physical shapes. For such solutions, parallels can be drawn with some of the accommodation structures seen in exposed escarpments of real geological folds.

Customized Coarse Models for Highly Heterogeneous Materials

This paper demonstrates the huge computational gains achieved when using a novel mulffitilevel Monte Carlo methodology for a typical aerospace model problem. To demonstrate the gains, we quantify the structural performance of a composite wing skin panel with three types of random manufacturing defects (ply angle perturbations and two classes of localised fibre waviness). We introduce the multilevel Monte Carlo method in an abstract way so that it could easily be applied also to other similar problems. We theoretically com- pare its complexity to standard Monte Carlo simulation and provide a simple-to-implement practical algorithm. Numerical experiments for all types of defects in our model problem confirm the theoretically predicted gains with as much as 152-fold computational speed-ups, bringing problems that would otherwise be unthinkable into the feasible range.