An efficient numerical integration system for stiff unified constitutive equations for metal forming applications

Unified constitutive equations have been developed in recent years to predict viscoplastic flow and microstructural evolution of metal alloys for metal forming applications. These equations can be implemented into commercial FE code, such as ABAQUS and PAMSTAMP, to predict mechanical and physical properties of materials in a wide range of metal forming processes. These equations are normally stiff and need significant computer CPU time to solve. In this research, a series of numerical analyses are performed to investigate the difficulties within MATLAB of solving these stiff unified constitutive equations. A metric is introduced to allow evaluation of the numerical stiffness to assess the most appropriate numerical integration method. This metric is based on the ratio of maximum to minimum eigenvalue. This metric allows for an appropriate numerical method to be chosen giving more effective modelling of deformation and plasticity processes. Based on the theoretical work described above, a user-friendly system, based on MATLAB, is then developed for numerically integrating these types of stiff constitutive equations. This is particularly useful for metal forming engineers and researchers who need an effective computational tool to determine constitutive properties well based on numerical integration theories

A conservative implicit multirate method for hyperbolic problems

This work focuses on the development of a self adjusting multirate strategy based on an implicit time discretization for the numerical solution of hyperbolic equations, that could benefit from different time steps in different areas of the spatial domain. We propose a novel mass conservative multirate approach, that can be generalized to various implicit time discretization methods. It is based on flux partitioning, so that flux exchanges between a cell and its neighbors are balanced. A number of numerical experiments on both non-linear scalar problems and systems of hyperbolic equations have been carried out to test the efficiency and accuracy of the proposed approach

Numerical approximation of a mechanochemical interface model for skin appendage

We introduce a model for the mass transfer of molecular activators and inhibitors in two media separated by an interface, and study its interaction with the deformations exhibited by the two-layer skin tissue where they occur. The mathematical model results in a system of nonlinear advection-diffusion-reaction equations including cross-diffusion, and coupled with an interface elasticity problem. We propose a Galerkin method for the discretisation of the set of governing equations, involving also a suitable Newton linearisation, partitioned techniques, non-overlapping Schwarz alternating schemes, and high-order adaptive time stepping algorithms. The experimental accuracy and robustness of the proposed partitioned numerical methods is assessed, and some illustrating tests in 2D and 3D are provided to exemplify the coupling effects between the mechanical properties and the advection-diffusion-reaction interactions involving the two separate layers

An efficient IMEX-DG solver for the compressible Navier-Stokes equations for non-ideal gases

We propose an efficient, accurate and robust IMEX solver for the compressible Navier-Stokes equation describing non-ideal gases with general equations of state. The method, which is based on an $h-$adaptive Discontinuos Galerkin spatial discretization and on an Additive Runge Kutta IMEX method for time discretization, is tailored for low Mach number applications and allows to simulate low Mach regimes at a significantly reduced computational cost, while maintaining full second order accuracy also for higher Mach number regimes. The method has been implemented in the framework of the $deal.II$ numerical library, whose adaptive mesh refinement capabilities are employed to enhance efficiency. Refinement indicators appropriate for real gas phenomena have been introduced. A number of numerical experiments on classical benchmarks for compressible flows and their extension to real gases demonstrate the properties of the proposed method

Unconditionally Strong Stability Preserving Extensions of the TR-BDF2 Method

We analyze monotonicity, strong stability and positivity of the TR-BDF2 method, interpreting these properties in the framework of absolute monotonicity. The radius of absolute monotonicity is computed and it is shown that the parameter value which makes the method L-stable is also the value which maximizes the radius of monotonicity. In order to achieve unconditional monotonicity, hybrid variants of TR-BDF2 are proposed, that reduce the formal order of accuracy, while keeping the native L-stability property, which is useful for the application to stiff problems. Numerical experiments compare these different hybridization strategies to other methods used in stiff and mildly stiff problems. The results show that the proposed strategies provide a good compromise between accuracy and robustness at high CFL numbers, without suffering from the limitations of alternative approaches already available in literature