High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics

We present a class of hybridizable discontinuous Galerkin (HDG) methods for the numerical simulation of wave phenomena in acoustics and elastodynamics. The methods are fully implicit and high-order accurate in both space and time, yet computationally attractive owing to their following distinctive features. First, they reduce the globally coupled unknowns to the approximate trace of the velocity, which is defined on the element faces and single-valued, thereby leading to a significant saving in the computational cost. In addition, all the approximate variables (including the approximate velocity and gradient) converge with the optimal order of k + 1 in the L²-norm, when polynomials of degree k ? 0 are used to represent the numerical solution and when the time-stepping method is accurate with order k + 1. When the time-stepping method is of order k + 2, superconvergence properties allows us, by means of local postprocessing, to obtain better, yet inexpensive approximations of the displacement and velocity at any time levels for which an enhanced accuracy is required. In particular, the new approximations converge with order k + 2 in the L²-norm when k ? 1 for both acoustics and elastodynamics. Extensive numerical results are provided to illustrate these distinctive features.     

Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem

In this paper, we propose a method to improve the convergence rate of the lowest order Raviart-Thomas mixed finite element approximations for the second order elliptic eigenvalue problem. Here, we prove a supercloseness result for the eigenfunction approximations and use a type of finite element postprocessing operator to construct an auxiliary source problem. Then solving the auxiliary additional source problem on an augmented mixed finite element space constructed by refining the mesh or by using the same mesh but increasing the order of corresponding mixed finite element space, we can increase the convergence order of the eigenpair approximation. This postprocessing method costs less computation than solving the eigenvalue problem on the finer mesh directly. Some numerical results are used to confirm the theoretical analysis.     

Raviart-Thomas混合元的超收敛

考虑二阶椭圆方程Dirichlet边值问题在正则矩形网格上k阶RaviartThomas混合有限元的超收敛.对有限元解经插值处理后,与通常的有限元最优误差估计相比,收敛速度提高了两阶.     

A NEW APPROACH TO RECOVERY OF DISCONTINUOUS GALERKIN

A new recovery operator P ：Qn^disc（T）→Qn＋1^disc（M） for discontinuous Galerkin is derived. It is based on the idea of projecting a discontinuous, piecewise polynomial solution on a given mesh T into a higher order polynomial space on a macro mesh M. In order to do so, we define local degrees of freedom using polynomial moments and provide global degrees of freedom on the macro mesh. We prove consistency with respect to the local L2-projection, stability results in several norms and optimal anisotropic error estimates. As an example, we apply this new recovery technique to a stabilized solution of a singularly perturbed convection-diffusion problem using bilinear elements.     

Nonlinear averaging reconstruction method for phase-cycle SSFP

The ability to obtain high-quality images of small structures, such as the nerves of the inner ear, is important for the early diagnosis of numerous conditions. Balanced steady-state free precession (SSFP; e.g., true fast imaging with steady-state precession) is a fast acquisition method, but its use has been limited by the presence of off-resonance banding artifacts. To reduce these artifacts multiacquisition balanced SSFP with phase cycling is used, yielding multiple data sets in which the banding artifacts are spatially shifted with respect to each other (e.g., as in CISS). We present a new method, called nonlinear averaging (NLA), for combining these data sets to reduce banding artifacts. The NLA method arithmetically averages the three highest magnitude signals from four-phase-cycle SSFP data on a pixel-by-pixel basis. Simulations indicate that NLA offers improved signal-to-noise ratio (SNR) over the more standard maximum intensity projection (MIP) reconstruction. NLA is compared to MIP in simulations and volunteer tests. Simulations suggest that NLA provides substantially improved SNR compared to MIP. In a randomized blinded comparison of 10 volunteer studies, two radiologists found that NLA, compared to MIP, gave improved results. NLA also provided superior noise reduction and enhanced edge sharpness compared to MIP. We demonstrate that NLA, similar to MIP, improves SNR and image quality. It does so consistently in all situations to which it is applied.     

A semiautomatic postprocessing of liver R2* measurement for assessment of liver iron overload

Purpose

The purpose was to propose and evaluate a semiautomatic postprocessing method to measure liver R2^? values in patients with a broad range of liver iron content.

Materials and Methods

Multiecho gradient echo magnetic resonance images were acquired in patients diagnosed with thalassemia or other types of congenital anemias. Liver R2^? values were measured using a routine manually defined region-of-interest (mROI) method and a semiautomatic (SA) method. In the semiautomatic method, pixelwise (pSA) and averaged (aSA) signal fitting was performed on the segmented liver tissues after hepatic vessel extraction. The pixelwise fitting approach resulted in a liver R2^? map with an overlay of nonfitted pixels associated with noise performance. The following aSA approach derived overall R2^? by fitting the averaged signal intensities of all pixels within the liver ROI excluding vessels and nonfitted pixels. The measurement accuracy and interobserver agreement using mROI and the two semiautomatic approaches (pSA and aSA) were evaluated.

Results

In a total of 45 exams with R2^? ranging from 30 to 1500 s⁻¹, the R2^? measurements using all three methods were overall highly correlated and concordant with each other. R2^? values measured by aSA were consistently higher than those measured by mROI. At lower R2^? (<1000 s⁻¹), R2^? values measured by pSA were consistent with aSA but higher than mROI; with increasing R2^?, the pSA method became less stable and underestimated R2^? due to increased noise level. The interobserver agreement was higher for the aSA method compared to pSA and mROI.

Conclusion

The semiautomatic postprocessing method provides a promising tool for reliable liver R2^? measurement with additional information for overall evaluation of iron distribution and measurement confidence. This method may offer the potential of reducing interoperator variability and improving diagnostic confidence in patients with liver iron overload.     

Comparing anisotropic diffusion filters for the enhancement of sodium magnetic resonance images

The anisotropic diffusion (AND) filter, an image processing technique derived from physics, was applied to low-resolution sodium magnetic resonance imaging (MRI) to examine the possibilities of image enhancement by postprocessing. We compared six different variants of AND filters. Besides the qualitative good results on phantom measurements, quantitative analyses on MRI of human kidney yielded major improvements in noise reduction and other quality measures: the noise (i.e., the standard deviation in the image background) could be reduced to 1%-2% of its original value, while linear filters (Gaussian, Fermi, Hamming) achieved a reduction to 42%-64%. Besides that, less than 5% of structures and intensities are lost when using AND filters. Comparing the different variants, the two-dimensional and the three-dimensional AND filter outperformed the histogram-of-gradient and tensor-based AND filter. We envision that by using these AND filters, quantitative analysis of sodium MRI of kidney could be improved.     

A splitting technique of higher order for the Navier–Stokes equations

This article presents a splitting technique for solving the time dependent incompressible Navier–Stokes equations. Using nested finite element spaces which can be interpreted as a postprocessing step the splitting method is of more than second order accuracy in time. The integration of adaptive methods in space and time in the splitting are discussed. In this algorithm, a gradient recovery technique is used to compute boundary conditions for the pressure and to achieve a higher convergence order for the gradient at different points of the algorithm. Results on the ‘Flow around a cylinder’s- and the ‘Driven Cavity’s-problem are presented.     

二维Maxwell方程组的混合有限元高精度近似

该文研究二维Maxwell方程组的混合有限元高精度近似.在均匀矩形网格上, 采用一阶Nedelec混合元空间, 有限元解经三次投影插值后, 在L\+2范数意义下, 其收敛于精确解的速度由O(h\+2)提高至O(h\+4).     

非线性抛物积分微分方程的各向异性有限元高精度分析

本文讨论非线性抛物积分微分方程的各向异性有限元方法.在不引入真解的H1-Volterra投影的情况下得到了半离散格式下的整体超收敛.