Convergence of summation-by-parts finite difference methods for the wave equation

In this paper, we consider finite difference approximations of the second order wave equation. We use finite difference operators satisfying the summation-by-parts property to discretize the equation in space. Boundary conditions and grid interface conditions are imposed by the simultaneous-approximation-term technique. Typically, the truncation error is larger at the grid points near a boundary or grid interface than that in the interior. Normal mode analysis can be used to analyze how the large truncation error affects the convergence rate of the underlying stable numerical scheme. If the semi-discretized equation satisfies a determinant condition, two orders are gained from the large truncation error. However, many interesting second order equations do not satisfy the determinant condition. We then carefully analyze the solution of the boundary system to derive a sharp estimate for the error in the solution and acquire the gain in convergence rate. The result shows that stability does not automatically yield a gain of two orders in convergence rate. The accuracy analysis is verified by numerical experiments.Comment: In version 2, we have added a new section on the convergence analysis of the Neumann problem, and have improved formulations in many place

Microstructural characterisation and mechanical testing of zirconium alloys and hydrides

Microstructure and formation mechanisms of δ-hydrides in rolled and recrystallised fine grain (grain size ~11 μm) and ‘blocky alpha’ large grain (grain size >200 μm) Zircaloy-4 have been studied using electron backscatter diffraction (EBSD). A unique {0001}α ||{111}δ;α ||δ matrix-hydride orientation relationship (OR) was found which supports the hydride formation mechanism through partial dislocation gliding on alternate matrix basal planes. Hydride stringers in the fine grain material were found to form through sympathetic hydride growth across grain boundaries. The macroscopic alignment of hydride stringers perpendicular to the plate normal direction (ND) is due mainly to the matrix material texture while relates weakly to the grain boundary characters. Micropillar compression tests reveal that the properties of basal slip in Zircaloy-4 vary strongly with temperature between 298 K and 623 K and the change in critical resolved shear stress (CRSS) with temperature follows a thermally activated constitutive law. At 623 K, the addition of hydrogen solutes has led to more homogeneous basal slip and negligible strain hardening. The interactions between crystal slip and hydrides in Zircaloy-4 have been studied using micropillar compression tests of single crystal specimens (where hydrides are intragranular) and macroscale tensile tests of polycrystal specimens (where hydrides are intergranular). Results suggest that local shear along some of the hydride-matrix interfaces is favoured over slip in the matrix nearby, causing localised deformation. Slip bands in the matrix, when reaching the hydride-matrix interfaces (nearly) perpendicularly, can either get arrested at the interfaces or result in shear within the hydride. Using high angular resolution EBSD (HR-EBSD), these complicated slip-hydride interactions have been observed to give rise to significant geometrically necessary dislocation (GND) pile-up along the hydride-matrix interface, indicating the risks of local damage accumulation and the tendency of hydrogen diffusion towards the dislocated area at high temperature triggering delayed hydride cracking (DHC).Open Acces

An Improved High Order Finite Difference Method for Non-conforming Grid Interfaces for the Wave Equation

This paper presents an extension of a recently developed high order finite difference method for the wave equation on a grid with non-conforming interfaces. The stability proof of the existing methods relies on the interpolation operators being norm-contracting, which is satisfied by the second and fourth order operators, but not by the sixth order operator. We construct new penalty terms to impose interface conditions such that the stability proof does not require the norm-contracting condition. As a consequence, the sixth order accurate scheme is also provably stable. Numerical experiments demonstrate the improved stability and accuracy property

Self-healable copolymers via van der Waals interactions

In these studies we developed fluoro-containing copolymers that for well-defined monomer molar ratios exhibit autonomous self-healing. This behavior is attributed to inter-digitated copolymer topologies facilitated by van der Waals (vdW) interactions. The main feature of vdW interactions is large polarizability (hydrophobicity) which in thermoplastic polymers leads to non-directional interactions between neighboring macromolecular segments. However, dipole-dipole directionality of the side groups can be enhanced by alternating/random copolymer compositions resulting from inter-chain vdW forces. Upon mechanical damage macromolecular segments will energetically favor original inter-chain conformations and return to original conformations, thus self-heal. Using chemical imaging (IR) and 1H NMR spectroscopy (NOESY, COSY) through-space and through-bond interactions were identified and used as input to molecular dynamic (MD) simulations in an effort to identify molecular events governing self-healing behavior

Self-healing copolymers via van der Waals interactions

Materials with build-in responsive components are outstanding candidates for the development of sustainable technologies. Last decade efforts have primarily focused in incorporating supramolecular chemistry and reversible covalent bonding in the development of self-healing polymers. This lecture will outline recent advances in self-healing polymers, with the primary focus on the recent advances in the development of commodity self-healable polymers. Inspired by plants, self-healing can be achieved by incorporating viscoelastic responses to their microstructures during their formation, thus enabling deformation upon mechanical damage to close a wound. This can be achieved by introducing multiphase-separated polymers composed of polycaprolactone, butanediol, and hexamethylene diisocyanate precursors copolymerized into a self-healing polymer.(1) The presence of micro-phase separated fibrous morphologies facilitate repeatable self-healing due to stable interfacial regions between the hard and soft segments of the copolymer, thus enabling of storage of entropic energy upon mechanical damage to be recovered during self-healing. This talk will provide the framework of van der Waals interactions in acrylic-based copolymers able to self-heal upon mechanical damage.(2) This behavior occurs when the monomer molar ratios are within a relatively narrow compositional range, forming reversible ‘key-and-lock’ interactions with preferentially alternating copolymer topologies. The unique self-healing behavior is attributed to favorable inter-chain van der Waals (vdW) forces manifested by the increased cohesive energy densities (CED) forming ‘key-and-lock’ inter-chain junctions, enabling multiple recovery upon mechanical damage without external intervention. The concept of redesigning commodity copolymers without elaborate chemical modifications will facilitate a platform for many technological opportunities and the development of new generations of sustainable copolymers with controlled chain topologies that survive repetitive damage-repair cycles. Please click Additional Files below to see the full abstract