Chang explicit method (2007) has been shown to be unconditionally stable for a linear elastic system and any instantaneous stiffness softening system while it is only conditionally stable for any instantaneous stiffness hardening system. Its coefficients of the difference equation for displacement increment are functions of initial tangent stiffness. Since Chang explicit method is unconditionally stable for a linear elastic system and any instantaneous stiffness softening system, its stability range can be enlarged if the initial tangent stiffness is enlarged by an amplification factor and then this amplified initial tangent stiffness is used to determine the coefficients. The detailed implementation of this scheme for Chang explicit method is presented and the feasibility of this scheme is verified.

Theoretical studies and numerical experiments suggest that unstructured high-ordermethods can provide solutions to otherwise intractable fluid flow problems within complexgeometries. However, it remains the case that existing high-order schemes are generallyless robust and more complex to implement than their low-order counterparts. These issues,in conjunction with difficulties generating high-order meshes, have limited the adoptionof high-order techniques in both academia (where the use of low-order schemes remainswidespread) and industry (where the use of low-order schemes is ubiquitous). In this shortreview, issues that have hitherto prevented the use of high-order methods amongst anon-specialist community are identified, and current efforts to overcome these issues arediscussed. Attention is focused on four areas, namely the generation of unstructuredhigh-order meshes, the development of simple and efficient time integration schemes, th edevelopment of robust and accurate shock capturing algorithms, and finally the developmentof high-order methods that are intuitive and simple to implement. With regards to thisfinal area, particular attention is focused on the recently proposed flux reconstructionapproach, which allows various well known high-order schemes (such as nodal discontinuousGalerkin methods and spectral difference methods) to be cast within a single unifyingframework. It should be noted that due to the experience of the authors the review isdirected somewhat towards aerodynamic applications and compressible flow. However, many ofthe discussions have a wider applicability. Moreover, the tone of the review is intendedto be generally accessible, such that an extended scientific community can gain insightinto factors currently pacing the adoption of unstructured high-order methods.

Dans cet article sont présentées des évolutions du modèle RL nécessaires pour larésolution de problèmes de dynamique rapide. Le modèle initial, basé sur le comportementthermomécanique de l’alliage couplé à l’équation de la chaleur, a été implémenté dans lecadre de la dynamique basses fréquences dans des travaux antérieurs. L’utilisation de cemodèle non-linéaire dans le cadre de la dynamique rapide (réponse de la structure à unimpact) conduit à des solutions non physiques ou à des divergences des algorithmesd’intégration temporelle. Ces comportements sont principalement dus aux fortesnon-linéarités des équations régissant les cinétiques de transformation entre les deuxphases du matériau (austénite/martensite). L’adjonction d’un terme lissant dans leséquations différentielles permet de faire converger les algorithmes sans perturber lecomportement du modèle initial. Ce terme, d’un point de vue physique, traduit le fait quela transformation de phase ne se fait pas de façon instantanée, mais à une vitesse depropagation donnée. Les résultats issus d’un calcul éléments-finis considérant unestructure bidimensionnelle soumise à un impact sont présentés.