Plenty of new two-dimensional materials including graphyne, graphdiyne, graphone, and graphane have been proposed and unveiled after the discovery of the "wonder material" graphene. Graphyne and ...graphdiyne are two-dimensional carbon allotropes of graphene with honeycomb structures. Graphone and graphane are hydrogenated derivatives of graphene. The advanced and unique properties of these new materials make them highly promising for applications in next generation nanoelectronics. Here, we briefly review their properties, including structural, mechanical, physical, and chemical properties, as well as their synthesis and applications in nanotechnology. Graphyne is better than graphene in directional electronic properties and charge carriers. With a band gap and magnetism, graphone and graphane show important applications in nanoelectronics and spintronics. Because these materials are close to graphene and will play important roles in carbon-based electronic devices, they deserve further, careful, and thorough studies for nanotechnology applications.
We present and analyze an entropy-stable semi-discretization of the Euler equations based on high-order summation-by-parts (SBP) operators. In particular, we consider general multidimensional SBP ...elements, building on and generalizing previous work with tensor–product discretizations. In the absence of dissipation, we prove that the semi-discrete scheme conserves entropy; significantly, this proof of nonlinear L2 stability does not rely on integral exactness. Furthermore, interior penalties can be incorporated into the discretization to ensure that the total (mathematical) entropy decreases monotonically, producing an entropy-stable scheme. SBP discretizations with curved elements remain accurate, conservative, and entropy stable provided the mapping Jacobian satisfies the discrete metric invariants; polynomial mappings at most one degree higher than the SBP operators automatically satisfy the metric invariants in two dimensions. In three-dimensions, we describe an elementwise optimization that leads to suitable Jacobians in the case of polynomial mappings. The properties of the semi-discrete scheme are verified and investigated using numerical experiments.
•Entropy-stable, SBP discretizations for non-tensor product, curved elements.•Polynomial mappings can be at most one degree higher than the SBP operators.•3D mapping Jacobian found by solving an elementwise optimization problem.•Numerical experiments verify entropy stability and accuracy.
The entropy conservative/stable staggered grid tensor-product algorithm of Parsani et al. 1 is extended to multidimensional SBP discretizations. The required SBP preserving interpolation operators ...are proven to exist under mild restrictions and the resulting algorithm is proven to be entropy conservative/stable as well as elementwise conservative. For 2-dimensional simplex elements, the staggered grid algorithm is shown to be more accurate and have a larger maximum time step restriction as compared to the collocated algorithm. The staggered algorithm significantly reduces the number of (computationally expensive) two-point flux evaluations, which is potentially important for both explicit and implicit time-marching schemes. Furthermore, the staggered algorithm requires fewer degrees of freedom for comparable accuracy, which has favorable implications for implicit time-marching schemes.
•Develop an entropy conservative/stable scheme for multi-D SBP operators.•Prove that prolongation/restriction operator pairs exist under mild assumptions.•The staggered algorithm is more accurate and reduces the DOFs.•The staggered algorithm allows a larger stable time step.
This work focuses on simultaneous approximation terms (SATs) for multidimensional summation-by-parts (SBP) discretizations of linear second-order partial differential equations with variable ...coefficients. Through the analysis of adjoint consistency and stability, we present several conditions on the SAT penalties for general operators, including those operators that do not have nodes on their boundary or do not correspond with a collocation discontinuous Galerkin method. Based on these conditions, we generalize the modified scheme of Bassi and Rebay and the symmetric interior penalty Galerkin method to SBP-SAT discretizations. Numerical experiments are carried out on unstructured grids with triangular elements to verify the theoretical results.
The main theme of this work is the development and application of entropy-stable discretizations. These discretizations inherit a nonlinear stability property from the continuous equations and they ...show great promise toward making the simulation of nonlinear hyperbolic partial differential equations as reliable as the simulation of linear partial differential equations. Furthermore, they also enable high-order discretizations to be applied without the stability limitations of many existing high-order methods. The first part of this thesis is the development of the entropy-stable discretization for use on unstructured grids and the proofs of its theoretical properties, including accuracy, stability, and discrete conservation. These properties are also extended to curved elements. A critical aspect of the discretizations is the use of a summation-by-parts operator, which has discrete properties that avoid assumptions on the smoothness of the solution or the accuracy of the quadrature rule in the stability proof. Several test problems are run to verify these properties and to compare against standard discretizations, showing the practical benefits of the theoretical developments. The second part of this thesis is a method to rapidly compute approximate functional values for mildly nonlinear problems. This method extends the theory of the adjoint to incorporate changes to both the solution and the geometry simultaneously. By using the solution on an initial geometry, the solution on the new geometry can be computed by solving the governing equations on a subset of the domain. The accurate prediction of which portions of the domain should be re-solved depends on the nonlinearity of the problem, and the proposed reanalysis method is accurate for problems where the strength of the nonlinearity is limited. The final part of this thesis is an $r$-adaptation method that uses the properties of the entropy-stable discretization to change the mesh node coordinates. This method uses an optimization problem to minimize the dissipation introduced by the discretization, which can be interpreted as reducing the magnitude of the modes that cannot be represented on the given mesh. The effect is that the mesh is aligned with flow features, leading to improvements in both solution and drag coefficient error for the example aerodynamic problems considered.
Entropy-conservative numerical flux functions can be used to construct high-order, entropy-stable discretizations of the Euler and Navier-Stokes equations. The purpose of this short communication is ...to present a novel family of such entropy-conservative flux functions. The proposed flux functions are solutions to quadratic optimization problems and admit closed-form, computationally affordable expressions. We establish the properties of the flux functions including their continuous differentiability, which is necessary for high-order discretizations.
This work focuses on multidimensional summation-by-parts (SBP) discretizations of linear elliptic operators with variable coefficients. We consider a general SBP discretization with dense ...simultaneous approximation terms (SATs), which serve as interior penalties to enforce boundary conditions and inter-element coupling in a weak sense. Through the analysis of adjoint consistency and stability, we present several conditions on the SAT penalties. Based on these conditions, we generalize the modified scheme of Bassi and Rebay (BR2) and the symmetric interior penalty Galerkin (SIPG) method to SBP-SAT discretizations. Numerical experiments are carried out on unstructured grids with triangular elements to verify the theoretical results.
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