a b s t r a c tNektar++ is an open-source software framework designed to support the development of highperformance scalable solvers for partial differential equations using the spectral/hp element method. High-order methods are gaining prominence in several engineering and biomedical applications due to their improved accuracy over low-order techniques at reduced computational cost for a given number of degrees of freedom. However, their proliferation is often limited by their complexity, which makes these methods challenging to implement and use. Nektar++ is an initiative to overcome this limitation by encapsulating the mathematical complexities of the underlying method within an efficient C++ framework, making the techniques more accessible to the broader scientific and industrial communities. The software supports a variety of discretisation techniques and implementation strategies, supporting methods research as well as application-focused computation, and the multi-layered structure of the framework allows the user to embrace as much or as little of the complexity as they need. The libraries capture the mathematical constructs of spectral/hp element methods, while the associated collection of pre-written PDE solvers provides out-of-the-box application-level functionality and a template for users who wish to develop solutions for addressing questions in their own scientific domains. Program summaryProgram title: Nektar++ Catalogue identifier: AEVV_v1_0Program summary URL:

No abstract

No abstract

An accurate calculation of aerodynamic force coefficients for a given geometry is of fundamental importance for aircraft design. High-order spectral/hp element methods, which use a discontinuous Galerkin discretisation of the compressible Navier-Stokes equations, are now increasingly being used to improve the accuracy of flow simulations and thus the force coefficients. To reduce error in the calculated force coefficients whilst keeping computational cost minimal, we propose a p-adaptation method where the degree of the approximating polynomial is locally increased in the regions of the flow where low resolution is identified using a goal-based error estimator as follows.Given an objective functional such as the aerodynamic force coefficients, we use control theory to derive an adjoint problem which provides the sensitivity of the functional with respect to changes in the flow variables, and assume that these changes are represented by the local truncation error. In its final form, the goal-based error indicator represents the effect of truncation error on the objective functional, suitably weighted by the adjoint solution. Both flow governing and adjoint equations are solved by the same high-order method, where we allow the degree of the polynomial within an element to vary across the mesh.We initially calculate a steady-state solution to the governing equations using resolve the force coefficients to a given error, as well as the computational cost, are both observed in using the p-adaptive technique.

The design of a modular multi-physics high-order space-time finite-element framework is presented together with its extension to allow monolithic coupling of different physics. One of the main objectives of the framework is to perform efficient high-fidelity simulations of capsule/parachute systems. This problem requires simulating multiple physics including, but not limited to, the compressible Navier-Stokes equations, the dynamics of a moving body with mesh deformations and adaptation, the linear shell equations, non-reflective boundary conditions and wall modeling. The solver is based on high-order space-time finite element methods. Continuous, discontinuous and C 1 -discontinuous Galerkin methods are implemented, allowing one to discretize various physical models. Tangent and adjoint sensitivity analysis are also targeted in order to conduct gradient-based optimization, error estimation, mesh adaptation, and flow control, adding another layer of complexity to the framework. The decisions made to tackle these challenges are presented. The discussion focuses first on the "single-physics" solver and later on its extension to the monolithic coupling of different physics. The implementation of different physics modules, relevant to the capsule/parachute system, are also presented. Finally, examples of coupled computations are presented, paving the way to the simulation of the full capsule/parachute system. * USRA/NASA Postdoctoral Program Fellow. † Science and Technology Corp. ‡ NASA ARC. AIAA Member.