scholarly journals Global regularity for the Monge-Ampère equation with natural boundary condition

2021 ◽  
Vol 194 (3) ◽  
pp. 745-793
Author(s):  
Shibing Chen ◽  
Jiakun Liu ◽  
Xu-Jia Wang
Keyword(s):  
Boundary Condition ◽  
Global Regularity ◽  
Natural Boundary Condition ◽  
Natural Boundary ◽  
Monge Ampere

Local Damage Evolution of Double Cantilever Beam Specimens During Crack Initiation Process: A Natural Boundary Condition Based Method

2009 ◽  
Vol 76 (5) ◽  
Author(s):  
Zhenyu Ouyang ◽  
Guoqiang Li
Keyword(s):  
Boundary Condition ◽  
Crack Initiation ◽  
Current Model ◽  
Crack Tip ◽  
Local Deformation ◽  
Interfacial Stress ◽  
Natural Boundary Condition ◽  
Natural Boundary ◽  
Initiation Process ◽  
Cohesive Laws

Cohesive zone models are being increasingly used to simulate fracture and debonding processes in metallic, polymeric, and ceramic materials and their composites. The crack initiation process as well as its actual stress and damage distribution beyond crack tip are important for understanding fracture of materials and debonding of adhesively bonded joints. In the current model, a natural boundary condition based method is proposed, and thus the concept of extended crack length (characteristic length l) is no longer required and more realistic and natural local deformation beyond crack tip can be obtained. The new analytical approach, which can consider both crack initiation and propagation as well as local deformation and interfacial stress distribution, can be explicitly obtained as a function of the remote peel load P with the given bilinear cohesive laws. An intrinsic geometric constraint condition is then used to solve the remote peel load P. The nonlinear response in both the ascending and descending stages of loading is accurately predicted by the current model, as evidenced by a comparison with both experimental results and finite element analysis results. It is found that the local deformation and interfacial stress beyond crack tip are relatively stable during crack propagation. It is also found that, when the cohesive strength is low, it has a significant effect on the critical peel load and loadline deflection. In principle, the approach developed in the current study can be extended to multilinear cohesive laws, although only bilinear cohesive law is presented in this work as an example.

scholarly journals Natural boundary condition methods for nuclear reactions

1976 ◽  
Vol 257 (3) ◽  
pp. 378-388 ◽  
Author(s):  
S.S. Ahmad ◽  
R.F. Barrett ◽  
B.A. Robson
Keyword(s):  
Boundary Condition ◽  
Nuclear Reactions ◽  
Natural Boundary Condition ◽  
Natural Boundary

Natural boundary condition methods for nuclear reactions (II)

1976 ◽  
Vol 270 (1) ◽  
pp. 1-14 ◽  
Author(s):  
S.S. Ahmad ◽  
R.F. Barrett ◽  
B.A. Robson
Keyword(s):  
Boundary Condition ◽  
Nuclear Reactions ◽  
Natural Boundary Condition ◽  
Natural Boundary

scholarly journals Incorporating arbitrary basal topography in the variational formulation of ice-sheet models

2011 ◽  
Vol 57 (203) ◽  
pp. 461-467 ◽  
Author(s):  
John K. Dukowicz ◽  
Stephen F. Price ◽  
William H. Lipscomb
Keyword(s):  
Boundary Condition ◽  
Variational Principle ◽  
Boundary Conditions ◽  
Variational Formulation ◽  
Ice Sheet ◽  
Approximate Model ◽  
Dirichlet Condition ◽  
Natural Boundary Condition ◽  
Natural Boundary ◽  
Basal Sliding

AbstractThere are many advantages to formulating an ice-sheet model in terms of a variational principle. In particular, this applies to the specification of boundary conditions, which might otherwise be problematic to implement. Here we focus primarily on the frictional basal sliding boundary condition in a non-Newtonian Stokes model. This type of boundary condition is particularly difficult because it is heterogeneous, requiring both a Dirichlet (no-penetration) condition normal to the bed, and a Neumann (frictional sliding) condition tangential to the bed. In general, Neumann conditions correspond to natural boundary conditions in a variational principle; that is, they arise naturally in the variational formulation and thus need not be explicitly specified. While the same is not necessarily true of Dirichlet conditions, it is possible to enforce a no-penetration condition using Lagrange multipliers within the variational principle so that the Dirichlet condition becomes a natural boundary condition. Thus, in the case of ice sheets, all relevant boundary conditions may be incorporated in the variational functional, making them particularly easy to discretize. For the Stokes model, the resulting basal boundary condition is valid for arbitrary topographic slopes. Here we apply the same methodology to the Blatter– Pattyn higher-order approximate model, which is ordinarily limited to small basal slopes by the smallaspect-ratio approximation. We introduce a modification that improves on the accuracy of the standard Blatter–Pattyn model for all values of the basal slope, as we demonstrate in the slow sliding regime for which analytical results are available. The remaining error is due to the effects of the small-aspect-ratio approximation in the Blatter–Pattyn model.

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