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When loading experiments are repeated on different samples, qualitatively different results can occur. This is due to factors such as geometric imperfections, load asymmetries, unevenly stressed regions or uneven material distributions created by manufacturing processes. This fact makes designing robust thin-walled structures difficult. One numerical strategy for exploring these different possible responses is to impose various initial imperfections on the geometry before loading, leading to different final solutions. However, this strategy is tedious, error-prone and gives an incomplete picture of the possible buckled configurations of the system. The present study demonstrates how a deflation strategy can be applied to obtain multiple solutions for the more robust design of thin-walled structures under displacement controlled uniaxial compression. We first demonstrate that distinct initial imperfections trigger different sequences of instability events in the Saint Venant–Kirchhoff hyperelastic model. We then employ deflation to investigate multiple bifurcation paths without the use of initial imperfections. A key advantage of this approach is that it can capture disconnected branches that cannot easily be discovered by conventional arc-length continuation and branch switching algorithms. Numerical experiments are given for three types of aircraft stiffener profiles. Our proposed technique is shown to be a powerful tool for exploring multiple disconnected bifurcation paths without requiring detailed insight for designing initial imperfections. We hypothesise that this technique will be very useful in the design process of robust thin-walled structures that must consider a variety of bifurcation paths., Aerospace Structures & Computational Mechanics

1. Humanity is on a deeply unsustainable trajectory. We are exceeding planetary boundaries and unlikely to meet many international sustainable development goals and global environmental targets. Until recently, there was no broadly accepted framework of interventions that could ignite the transformations needed to achieve these desired targets and goals. 2. As a component of the IPBES Global Assessment, we conducted an iterative expert deliberation process with an extensive review of scenarios and pathways to sustainability, including the broader literature on indirect drivers, social change and sustainability transformation. We asked, what are the most important elements of pathways to sustainability? 3. Applying a social-ecological systems lens, we identified eight priority points for intervention (leverage points) and five overarching strategic actions and priority interventions (levers), which appear to be key to societal transformation. The eight leverage points are: (1) Visions of a good life, (2) Total consumption and waste, (3) Latent values of responsibility, (4) Inequalities, (5) Justice and inclusion in conservation, (6) Externalities from trade and other telecouplings, (7) Responsible technology, innovation and investment, and (8) Education and knowledge generation and sharing. The five intertwined levers can be applied across the eight leverage points and more broadly. These include: (A) Incentives and capacity building, (B) Coordination across sectors and jurisdictions, (C) Pre-emptive action, (D) Adaptive decision-making and (E) Environmental law and implementation. The levers and leverage points are all non-substitutable, and each enables others, likely leading to synergistic benefits. 4. Transformative change towards sustainable pathways requires more than a simple scaling-up of sustainability initiatives-it entails addressing these levers and leverage points to change the fabric of legal, political, economic and other social systems. These levers a

Augmented Lagrangian preconditioners have successfully yielded Reynolds-robust preconditioners for the stationary incompressible Navier-Stokes equations, but only for specific discretizations. The discretizations for which these preconditioners have been designed possess error estimates which depend on the Reynolds number, with the discretization error deteriorating as the Reynolds number is increased. In this paper we present an augmented Lagrangian preconditioner for the Scott-Vogelius discretization on barycentrically-refined meshes. This achieves both Reynolds-robust performance and Reynolds-robust error estimates. A key consideration is the design of a suitable space decomposition that captures the kernel of the grad-div term added to control the Schur complement; the same barycentric refinement that guarantees inf-sup stability also provides a local decomposition of the kernel of the divergence. The robustness of the scheme is confirmed by numerical experiments in two and three dimensions., Comment: Fixed sign of grad-div term in (3.7)

We propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen-Frank model arising in cholesteric liquid crystals. By applying the augmented Lagrangian method, the Schur complement of the director block can be better approximated by the weighted mass matrix of the Lagrange multiplier, at the cost of making the augmented director block harder to solve. In order to solve the augmented director block, we develop a robust multigrid algorithm which includes an additive Schwarz relaxation that captures a pointwise version of the kernel of the semi-definite term. Furthermore, we prove that the augmented Lagrangian term improves the discrete enforcement of the unit-length constraint. Numerical experiments verify the efficiency of the algorithm and its robustness with respect to problem-related parameters (Frank constants and cholesteric pitch) and the mesh size.

We present a mesh-independent and parameter-robust multigrid solver for the Scott-Vogelius discretisation of the nearly incompressible linear elasticity equations on meshes with a macro element structure. The discretisation achieves exact representation of the limiting divergence constraint at moderate polynomial degree. Both the relaxation and multigrid transfer operators exploit the macro structure for robustness and efficiency. For the relaxation, we use the existence of local Fortin operators on each macro cell to construct a local space decomposition with parameter-robust convergence. For the transfer, we construct a robust prolongation operator by performing small local solves over each coarse macro cell. The necessity of both components of the algorithm is confirmed by numerical experiments.

We devise 3-field and 4-field finite element approximations of a system describing the steady state of an incompressible heat-conducting fluid with implicit non-Newtonian rheology. We prove that the sequence of numerical approximations converges to a weak solution of the problem. We develop a block preconditioner based on augmented Lagrangian stabilisation for a discretisation based on the Scott-Vogelius finite element pair for the velocity and pressure. The preconditioner involves a specialised multigrid algorithm that makes use of a space-decomposition that captures the kernel of the divergence and non-standard intergrid transfer operators. The preconditioner exhibits robust convergence behaviour when applied to the Navier-Stokes and power-law systems, including temperature-dependent viscosity, heat conductivity and viscous dissipation., Comment: 9 figures, 39 pages