Accuracy analysis of gradient reconstruction on isotropic unstructured meshes and its effects on inviscid flow simulation

The accuracy of gradient reconstruction methods on unstructured meshes is analyzed both mathematically and numerically.Mathematical derivations reveal that,for gradient reconstruction based on the Green-Gauss theorem(the GG methods),if the summation of first-and-lower-order terms does not counterbalance in the discretized integral process,which rarely occurs,second-order accurate approximation of face midpoint value is necessary to produce at least first-order accurate gradient.However,gradient reconstruction based on the least-squares approach(the LSQ methods)is at least first-order on arbitrary unstructured grids.Verifications are performed on typical isotropic grid stencils by analyzing the relationship between the discretization error of gradient reconstruction and the discretization error of the face midpoint value approximation of a given analytic function.Meanwhile,the numerical accuracy of gradient reconstruction methods is examined with grid convergence study on typical isotropic grids.Results verify the phenomenon of accuracy degradation for the GG methods when the face midpoint value condition is not satisfied.The LSQ methods are proved to be at least first-order on all tested isotropic grids.To study gradient accuracy effects on inviscid flow simulation,solution errors are quantified using the Method of Manufactured Solutions(MMS)which was validated before adoption by comparing with an exact solution case,i.e.,the 2-dimensional(2D)inviscid isentropic vortex.Numerical results demonstrate that the order of accuracy(OOA)of gradient reconstruction is crucial in determining the OOA of numerical solutions.Solution accuracy deteriorates seriously if gradient reconstruction does not reach first-order.

Optimization-based image reconstruction in x-ray computed tomography by sparsity exploitation of local continuity and nonlocal spatial self-similarity

The additional sparse prior of images has been the subject of much research in problems of sparse-view computed tomography（CT） reconstruction. A method employing the image gradient sparsity is often used to reduce the sampling rate and is shown to remove the unwanted artifacts while preserve sharp edges, but may cause blocky or patchy artifacts.To eliminate this drawback, we propose a novel sparsity exploitation-based model for CT image reconstruction. In the presented model, the sparse representation and sparsity exploitation of both gradient and nonlocal gradient are investigated.The new model is shown to offer the potential for better results by introducing a similarity prior information of the image structure. Then, an effective alternating direction minimization algorithm is developed to optimize the objective function with a robust convergence result. Qualitative and quantitative evaluations have been carried out both on the simulation and real data in terms of accuracy and resolution properties. The results indicate that the proposed method can be applied for achieving better image-quality potential with the theoretically expected detailed feature preservation.

Applications of multi-dimensional schemes on unstructured grids for high-accuracy heat flux prediction

The simulation of hypersonic flows with fully unstructured(tetrahedral)grids has severe problems with respect to the prediction of stagnation region heating,due to the random face orientation without alignment to the bow shock.To improve the accuracy of aero-heating predictions,three multi-dimensional approaches on unstructured grids are coupled in our Reynolds-averaged Navier-Stokes(RANS)solver,including multi-dimensional upwind flux reconstruction(MUP),multi-dimensional limiter(MLP-u2)and multi-dimensional gradient reconstruction(MLR).The coupled multi-dimensional RANS solver is validated by several typical verification and validation(V&V)cases,including hypersonic flows over a cylinder,a blunt biconic,and a double-ellipsoid,with commonly used prism/tetrahedral hybrid grids.Finally,the coupled multi-dimensional solver is applied to simulating the heat flux distribution over a 3D engineering configuration,i.e.a Hermes-like space shuttle model.The obtained numerical results are compared with experimental data.The predicted results demonstrate that the coupled multi-dimensional approach has a good prediction capability on aerodynamic heating over a wide range of complex engineering configurations.