Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains

This article is concerned with the discretisation of the Stokes equations on time-dependent domains in an Eulerian coordinate framework. Our work can be seen as an extension of a recent paper by Lehrenfeld and Olshanskii (ESAIM: M2AN 53(2):585–614, 2019), where BDF-type time-stepping schemes are studied for a parabolic equation on moving domains. For space discretisation, a geometrically unfitted finite element discretisation is applied in combination with Nitsche’s method to impose boundary conditions. Physically undefined values of the solution at previous time-steps are extended implicitly by means of so-called ghost penalty stabilisations. We derive a complete a priori error analysis of the discretisation error in space and time, including optimal $$L^2(L^2)$$ L 2 ( L 2 ) -norm error bounds for the velocities. Finally, the theoretical results are substantiated with numerical examples.

Multi-scale hydro-morphodynamic modelling using mesh movement methods

Hydro-morphodynamic modelling is an important tool that can be used in the protection of coastal zones. The models can be required to resolve spatial scales ranging from sub-metre to hundreds of kilometres and are computationally expensive. In this work, we apply mesh movement methods to a depth-averaged hydro-morphodynamic model for the first time, in order to tackle both these issues. Mesh movement methods are particularly well-suited to coastal problems as they allow the mesh to move in response to evolving flow and morphology structures. This new capability is demonstrated using test cases that exhibit complex evolving bathymetries and have moving wet-dry interfaces. In order to be able to simulate sediment transport in wet-dry domains, a new conservative discretisation approach has been developed as part of this work, as well as a sediment slide mechanism. For all test cases, we demonstrate how mesh movement methods can be used to reduce discretisation error and computational cost. We also show that the optimum parameter choices in the mesh movement monitor functions are fairly predictable based upon the physical characteristics of the test case, facilitating the use of mesh movement methods on further problems.

Expansions for the linear-elastic contribution to the self-interaction force of dislocation curves

The self-interaction force of dislocation curves in metals depends on the local arrangement of the atoms and on the non-local interaction between dislocation curve segments. While these non-local segment–segment interactions can be accurately described by linear elasticity when the segments are further apart than the atomic scale of size $\varepsilon$ , this model breaks down and blows up when the segments are $O(\varepsilon)$ apart. To separate the non-local interactions from the local contribution, various models depending on $\varepsilon$ have been constructed to account for the non-local term. However, there are no quantitative comparisons available between these models. This paper makes such comparisons possible by expanding the self-interaction force in these models in $\varepsilon$ beyond the O(1)-term. Our derivation of these expansions relies on asymptotic analysis. The practical use of these expansions is demonstrated by developing numerical schemes for them, and by – for the first time – bounding the corresponding discretisation error.

Upscaled exact solutions to root water uptake equations for earth system modelling

<p>Earth system models struggle to accurately predict soil-root water flows, especially under drying or heterogeneous soil moisture conditions, resulting in inaccurate description of water limitation of terrestrial fluxes. Recent descriptions of plant hydraulics address this by applying Ohm’s law analogues to the soil-plant-atmosphere hydraulic continuum.</p><p>While adequate for stems, this formulation linearises soil-root and within-root resistances by assumption, neglecting the nonlinearity of pressure gradients in absorbing roots. The resulting discretisation error is known to depend strongly on model spatial resolution. At coarse resolution, substantial errors arise in a form depending on the assumed configuration of resistances. In simulations of a drought at the Wind River Crane (WRC) flux site, a parallel Ohm model based on the rooting profile overpredicted hydraulic redistribution, while a series model overpredicted uptake in shallow layers at the expense of deep ones.</p><p>A conceptual alternative is to upscale exact solutions to the hyperbolic differential equation that describes root water uptake, by solving for the mean root water potential in each soil subdomain. Upscaled solutions show that multiple soil water potentials affect pressure gradients in each root segment, producing the nonlinearities absent in Ohm models. This upscaled model gave better predictions of WRC drought data and was significantly less prone to over-fitting than the two Ohm models, with more robust predictions beyond calibration conditions.</p><p>Analysis reveals classes of root systems of differing architectural complexity that yield a common upscaled model. In numerical experiments, using a simple upscaled model in situations of increasing complexity (e.g., adding individual plants), resulted in bounded errors that decreased asymptotically with increased complexity. The approach is thus a viable candidate for upscaling the effects of heterogenous soil moisture distributions on root water uptake.</p>

Application of gPCRK Methods to Nonlinear Random Differential Equations with Piecewise Constant Argument

We propose a class of numerical methods for solving nonlinear random differential equations with piecewise constant argument, called gPCRK methods as they combine generalised polynomial chaos with Runge-Kutta methods. An error analysis is presented involving the error arising from a finite-dimensional noise assumption, the projection error, the aliasing error and the discretisation error. A numerical example is given to illustrate the effectiveness of this approach.

Prosthetic foot design optimisation based on roll-over shape and ground reaction force characteristics

Most prosthetic foot products are adjusted by skilled prosthetists using a variety of gait analysis and information given to them by patients in terms of feel and experience. The design of prosthetic foot has traditionally focused on optimising stiffness to support the body weight and storage/release mechanisms of strain energy from heel contact to push off. As a result of this, the optimisation process of a prosthetic foot is simple and sometimes insufficient. It is proposed that the stiffness and energy release mechanisms of prosthetic feet be designed to satisfy amputee’s natural gait characteristics that are defined by an effective roll-over shape and corresponding ground reaction force combinations. Each point on the roll-over shape is mapped with a ground reaction force corresponding to its time stamp. The resulting discrete set of ground reaction force components are applied to the prosthetic foot sole and its stiffness profile is optimised to produce a desired deflection as given by the corresponding point on the roll-over shape. The robustness of this novel computational method is tested on three prosthetic designs. The mesh sensitivity results and the discretisation error resulting from applying finite number of ground reaction forces are discussed. It is shown that the proposed methodology is able to provide valuable insights in the guidelines for selection of materials for a multi-material prosthetic foot.

Experimental and Simulative Investigations of Tribology in Sheet-Bulk Metal Forming

Friction has a considerable influence in metal forming both in economic and technical terms. This is especially true for sheet-bulk metal forming (SBMF). The contact pressure that occurs here can be low making Coulomb’s friction law advisable, but also very high so that Tresca’s friction law is preferable. By means of an elasto-plastic half-space model rough surfaces have been investigated, which are deformed in such contact states. The elasto-plastic half-space model has been verified and calibrated experimentally. The result is the development of a constitutive friction law, which can reproduce the frictional interactions for both low and high contact pressures. In addition, the law gives conclusion regarding plastic smoothening of rough surfaces. The law is implemented in the framework of the Finite-Element-Method. However, compared to usual friction relations the tribological interplay presented here comes with the disadvantage of rising numerical effort. In order to minimise this drawback, a model adaptive finite-element-simulation is performed additionally. In this approach, contact regions are identified, where a conventional friction law is applicable, where the newly developed constitutive friction law should be used, or where frictional effects are negligible. The corresponding goal-oriented indicators are derived based on the “dual-weighted-residual” (DWR) method taking into account both the model and the discretisation error. This leads to an efficient simulation that applies the necessary friction law in dependence of contact complexity.

Sound Propagation Properties of the Discrete-Velocity Boltzmann Equation

As the numerical resolution is increased and the discretisation error decreases, the lattice Boltzmann method tends towards the discrete-velocity Boltzmann equation (DVBE). An expression for the propagation properties of plane sound waves is found for this equation. This expression is compared to similar ones from the Navier-Stokes and Burnett models, and is found to be closest to the latter. The anisotropy of sound propagation with the DVBE is examined using a two-dimensional velocity set. It is found that both the anisotropy and the deviation between the models is negligible if the Knudsen number is less than 1 by at least an order of magnitude.

High order optimal anisotropic mesh adaptation using hierarchical elements

Anisotropic mesh adaptation has made spectacular progress in the past few years. The introduction of the notion of a metric, directly linked to the interpolation error, has allowed to control the elongation of elements as well as the discretisation error. This approach is however essentially restricted to linear (P(1)) finite element solutions, though there exists some generalisations. A completely general approach leading to optimal meshes and this, for finite element solution of any degree, is still missing. This is precisely the goal of this work where we show how to estimate the error on a finite element solution of degree k using hierarchical basis for Lagrange finite element polynomials. We then show how to use this information to produce optimal anisotropic meshes in a sense that will be precised.