Variational arbitrary Lagrangian-Eulerian method

ALE methods;Finite Elements;Mesh adaption;Shape Optimization;Variational Methods

Thoutireddy, Pururav ; Ortiz, Michael

University:California Institute of Technology

Department:Engineering and Applied Science

关键词: ALE methods;    Finite Elements;    Mesh adaption;    Shape Optimization;    Variational Methods;   

Others  :  https://thesis.library.caltech.edu/2227/1/thesis.pdf

美国|英语

来源: Caltech THESIS

【 摘 要 】

This thesis is concerned with the development of Variational Arbitrary Lagrangian-Eulerian method (VALE) method. VALE is essentially finite element method generalized to account for horizontal variations, in particular, variations in nodal coordinates. The distinguishing characteristic of the method is that the variational principle simultaneously supplies the solution, the optimal mesh and, in case problems of shape optimization, optimal shape. This is accomplished by rendering the functional associated with the variational principle stationary with respect to nodal field values as well as with respect to the nodal positions of triangulation of the domain of analysis. Stationarity with respect to the nodal positions has the effect of the equilibriating the energetic or configurational forces acting in the nodes. Further, configurational force equilibrium provides precise criterion for mesh optimality. The solution so obtained corresponds to minimum of energy functional (minimum principle) in static case and to the stationarity of action sum (discrete Hamilton's stationarity principle) in dynamic case, with respect to both nodal variables and nodal positions. Further, the resulting mesh adaption scheme is devoid of error estimates and mesh-to-mesh transfer interpolation errors. We illustrate the versatility and convergence characteristics of the method by way of selected numerical tests and applications, including the problem of semi-infinite crack, the shape optimization of elastic inclusions and free vibration of 1-d rod.