On mesh restrictions to satisfy comparison principles,maximum principles,and the non-negative constraint: Recent developments and new results

This article concerns mesh restrictions that are needed to satisfy several important mathematical properties—maximum principles, comparison principles, and the nonnegative constraint—for a general linear second-order elliptic partial differential equation. We critically review some recent developments in the field of discrete maximum principles, derive new results, and discuss some possible future research directions in this area. In particular, we derive restrictions for a three-node triangular (T3) element and a four-node quadrilateral (Q4) element to satisfy comparison principles, maximum principles, and the nonnegative constraint under the standard single-field Galerkin formulation. Analysis is restricted to uniformly elliptic linear differential operators in divergence form with Dirichlet boundary conditions specified on the entire boundary of the domain. Various versions of maximum principles and comparison principles are discussed in both continuous and discrete settings. In the literature, it is well-known that an acute-angled triangle is sufficient to satisfy the discrete weak maximum principle for pure isotropic diffusion. Herein, we show that this condition can be either too restrictive or not sufficient to satisfy various discrete principles when one considers anisotropic diffusivity, advection velocity field, or linear reaction coefficient. Subsequently, we derive appropriate restrictions on the mesh for simplicial (e.g., T3 element) and nonsimplicial (e.g., Q4 element) elements. Based on these conditions, an iterative algorithm is developed to construct simplicial meshes that preserve discrete maximum principles using existing open source mesh generators. Various numerical examples based on different types of triangulations are presented to show the pros and cons of placing restrictions on a computational mesh. We also quantify local and global mass conservation errors using representative numerical examples and illustrate the performance of metric.     

Green intermodal freight transportation: bi-objective modelling and analysis

Efficient planning of freight transportation requires a comprehensive look at wide range of factors in the operation and management of any transportation mode to achieve safe, fast, and environmentally suitable movement of goods. In this regard, a combination of transportation modes offers flexible and environmentally friendly alternatives to transport high volumes of goods over long distances. In order to reflect the advantages of each transportation mode, it is the challenge to develop models and algorithms in Transport Management System software packages. This paper discusses the principles of green logistics required in designing such models and algorithms which truly represent multiple modes and their characteristics. Thus, this research provides a unique practical contribution to green logistics literature by advancing our understanding of the multi-objective planning in intermodal freight transportation. Analysis based on a case study from hinterland intermodal transportation in Europe is therefore intended to make contributions to the literature about the potential benefits from combining economic and environmental criteria in transportation planning. An insight derived from the experiments conducted shows that there is no need to greatly compromise on transportation costs in order to achieve a significant reduction in carbon-related emissions.