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Background Increased aortic stiffness is an independent predictor of cardiovascular disease. Optimal measurement is highly beneficial for the detection of atherosclerosis and the management of patients at risk. Thus, it was our purpose to selectively measure aortic stiffness using a novel imaging method and to provide reference values from a population-based study. Methods One hundred twenty six inhabitants of Freiburg, Germany, between 20 and 80 years prospectively underwent 3 Tesla cardiovascular magnetic resonance (CMR) of the thoracic aorta. 4D flow CMR (spatial/temporal resolution 2mm 3 /20ms) was executed to calculate aortic pulse wave velocity (PWV) in m/s using dedicated software. In addition, we calculated distensibility coefficients (DC) using 2D CINE CMR imaging of the ascending (AAo) and descending aorta (DAo). Segmental aortic diameter and thickness of aortic plaques were determined by 3D T1 weighted CMR (spatial resolution 1mm 3 ). Results PWV increased from 4.93 ± 0.54 m/s in 20–30 year-old to 8.06 ± 1.03 m/s in 70–80 year-old subjects. PWV was significantly lower in women compared to men ( p  < 0.0001). Increased blood pressure (systolic r  = 0.36, p  < 0.0001; diastolic r  = 0.33, p  = 0.0001; mean arterial pressure r  = 0.37, p  < 0.0001) correlated with PWV after adjustment for age and gender. Finally, PWV increased with increasing diameter of the aorta (ascending aorta r  = 0.20, p  = 0.026; aortic arch r  = 0.24, p  = 0.009; descending aorta r  = 0.26, p  = 0.004). Correlation of PWV and DC of the AAo and DAo or the mean of both was high ( r  = 0.69, r  = 0.68, r  = 0.73; p  < 0.001). Conclusions 4D flow CMR was successfully applied to calculate aortic PWV and thus aortic stiffness. Findings showed a high correlation with distensibility coefficients representing local compliance of the aorta. Our novel method and reference data for PWV may provide a reliable biomarker for the identification of patients with underlying cardiovascular disease and optimal guidance of future treatment in studies or clinical routine.

To quantify stiffness of the descending aorta (DAo) in stroke patients using 4D flow MRI and compare results with transesophageal echocardiography (TEE). 48 acute stroke patients undergoing 4D flow MRI and TEE were included. Intima-media-thickness (IMT) was measured in the DAo and the aorta was scrutinized for atherosclerotic plaques using TEE. Stiffness of the DAo was determined by (a) 4D flow MRI at 3 T by calculating pulse wave velocity (PWV) and by (b) TEE calculating arterial strain, stiffness index, and distensibility coefficient. Mean IMT was 1.43 ± 1.75. 7 (14.6%) subjects had no sign of atherosclerosis, 10 (20.8%) had IMT-thickening or plaques < 4 mm, and 31 (66.7%) had at least one large and/or complex plaque in the aorta. Increased IMT significantly correlated (p < 0.001) with increased DAo stiffness in MRI (PWV r = 0.66) and in TEE (strain r = 0.57, stiffness index r = 0.64, distensibility coefficient r = 0.57). Patients with at least IMT-thickening had significantly higher stiffness values compared to patients without atherosclerosis. However, no difference was observed between patients with plaques < 4 mm and patients with plaques ≥ 4 mm. PWV and TEE parameters of stiffness correlated significantly [strain (r = − 0.36; p = 0.011), stiffness index (r = 0.51; p = 0.002), and distensibility coefficient (r = − 0.59; p < 0.001)]. 4D flow MRI and TEE-based parameters of aortic stiffness were associated with markers of atherosclerosis such as IMT-thickness and presence of plaques. We believe that 4D flow MRI is a promising tool for future studies of aortic atherosclerosis, due to its longer coverage of the aorta and non-invasiveness.

The discontinuous Galerkin (DG) method has very quickly found utility in such diverse applications as computational solid mechanics, fluid mechanics, acoustics, and electromagnetics. The DG methodology merely requires weak constraints on the fluxes between elements. This feature provides a flexibility which is difficult to match with conventional continuous Galerkin methods. However, allowing discontinuity between element interfaces can in turn be problematic during simulation postprocessing, such as in visualization. Consequently, smoothness-increasing accuracy-conserving (SIAC) filters were proposed in [M. Steffen et al., IEEE Trans. Vis. Comput. Graph. , 14 (2008), pp. 680--692, D. Walfisch et al., J. Sci. Comput. , 38 (2009), pp. 164--184] as a means of introducing continuity at element interfaces while maintaining the order of accuracy of the original input DG solution. Although the DG methodology can be applied to arbitrary triangulations, the typical application of SIAC filters has been to DG solutions obtained over translation invariant meshes such as structured quadrilaterals and triangles. As the assumption of any sort of regularity including the translation invariance of the mesh is a hindrance towards making the SIAC filter applicable to real life simulations, we demonstrate in this paper for the first time the behavior and complexity of the computational extension of this filtering technique to fully unstructured tessellations. We consider different types of unstructured triangulations and show that it is indeed possible to get reduced errors and improved smoothness through a proper choice of kernel scaling. These results are promising, as they pave the way towards a more generalized SIAC filtering technique.

Theoretically and computationally, it is possible to demonstrate that the order of accuracy of a discontinuous Galerkin (DG) solution for linear hyperbolic equations can be improved from order k+1 to 2k+1 through the use of smoothness-increasing accuracy-conserving (SIAC) filtering. However, it is a computationally complex task to perform this in an efficient manner, which becomes an even greater issue considering nonquadrilateral mesh structures. In this paper, we present an extension of this SIAC filter to structured triangular meshes. The basic theoretical assumption in the previous implementations of the postprocessor limits the use to numerical solutions solved over a quadrilateral mesh. However, this assumption is restrictive, which in turn complicates the application of this postprocessing technique to general tessellations. Additionally, moving from quadrilateral meshes to triangulated ones introduces more complexity in the calculations as the number of integrations required increases. In this paper, we extend the current theoretical results to variable coefficient hyperbolic equations over structured triangular meshes and demonstrate the effectiveness of the application of this postprocessor to structured triangular meshes as well as exploring the effect of using inexact quadrature. We show that there is a direct theoretical extension to structured triangular meshes for hyperbolic equations with bounded variable coefficients. This is a challenging first step toward implementing SIAC filters for unstructured tessellations. We show that by using the usual B-spline implementation, we are able to improve on the order of accuracy as well as decrease the magnitude of the errors. These results are valid regardless of whether exact or inexact integration is used. The results here demonstrate that it is still possible, both theoretically and computationally, to improve to 2k+1 over the DG solution itself for structured triangular meshes.

The discontinuous Galerkin (DG) methods provide a high-order extension of the finite volume method in much the same way as high-order or spectral/ hp elements extend standard finite elements. However, lack of inter-element continuity is often contrary to the smoothness assumptions upon which many post-processing algorithms such as those used in visualization are based. Smoothness-increasing accuracy-conserving (SIAC) filters were proposed as a means of ameliorating the challenges introduced by the lack of regularity at element interfaces by eliminating the discontinuity between elements in a way that is consistent with the DG methodology; in particular, high-order accuracy is preserved and in many cases increased. The goal of this paper is to explicitly define the steps to efficient computation of this filtering technique as applied to both structured triangular and quadrilateral meshes. Furthermore, as the SIAC filter is a good candidate for parallelization, we provide, for the first time, results that confirm anticipated performance scaling when parallelized on a shared-memory multi-processor machine.

In this paper, we attempt to address the potential usefulness of smoothness-increasing accuracy-conserving (SIAC) filters when applied to real-world simulations. SIAC filters as a class of post-processors were initially developed in Bramble and Schatz (Math Comput 31:94, 1977 ) and later applied to discontinuous Galerkin (DG) solutions of linear hyperbolic partial differential equations by Cockburn et al. (Math Comput 72:577, 2003 ), and are successful in raising the order of accuracy from k + 1 to 2 k + 1 in the L 2 —norm when applied to a locally translation-invariant mesh. While there have been several attempts to demonstrate the usefulness of this filtering technique to nontrivial mesh structures (Curtis et al. in SIAM J Sci Comput 30(1):272, 2007 ; Mirzaee et al. in SIAM J Numer Anal 49:1899, 2011 ; King et al. in J Sci Comput, 2012 ), the application of the SIAC filter never exceeded beyond two-space dimensions. As tetrahedral meshes are often the type considered in more realistic simulations, we contribute to the class of SIAC post-processors by demonstrating the effectiveness of SIAC filtering when applied to structured tetrahedral meshes. These types of meshes are generated by tetrahedralizing uniform hexahedra and therefore, while maintaining the structured nature of a hexahedral mesh, they exhibit an unstructured tessellation within each hexahedral element. Moreover, we address the computationally intensive task of performing numerical integrations when one considers tetrahedral elements for SIAC filtering and provide guidelines on how to ameliorate these challenges through the use of more general cubature rules. We consider two examples of a hyperbolic equation and confirm the usefulness of SIAC filters in obtaining the superconvergence accuracy of 2 k + 1 when applied to structured tetrahedral meshes. Additionally, the DG methodology merely requires weak constraints on the fluxes between elements. As SIAC filters improve this weak continuity to C k - 1 —continuity at the element interfaces, we provide results that show how post-processing is useful in extracting smooth isosurfaces of DG fields.

Smoothness-increasing accuracy-conserving (SIAC) filtering has demonstrated its effectiveness in raising the convergence rate of discontinuous Galerkin solutions from order to order 2 k +1 for specific types of translation invariant meshes (Cockburn et al. in Math. Comput. 72:577–606, 2003 ; Curtis et al. in SIAM J. Sci. Comput. 30(1):272–289, 2007 ; Mirzaee et al. in SIAM J. Numer. Anal. 49:1899–1920, 2011 ). Additionally, it improves the weak continuity in the discontinuous Galerkin method to k −1 continuity. Typically this improvement has a positive impact on the error quantity in the sense that it also reduces the absolute errors. However, not enough emphasis has been placed on the difference between superconvergent accuracy and improved errors. This distinction is particularly important when it comes to understanding the interplay introduced through meshing, between geometry and filtering. The underlying mesh over which the DG solution is built is important because the tool used in SIAC filtering—convolution—is scaled by the geometric mesh size. This heavily contributes to the effectiveness of the post-processor. In this paper, we present a study of this mesh scaling and how it factors into the theoretical errors. To accomplish the large volume of post-processing necessary for this study, commodity streaming multiprocessors were used; we demonstrate for structured meshes up to a 50× speed up in the computational time over traditional CPU implementations of the SIAC filter.

The discontinuous Galerkin (DG) method continues to maintain heightened levels of interest within the simulation community because of the discretization flexibility it provides. Although one of the fundamental properties of the DG methodology and arguably its most powerful property is the ability to combine high-order discretizations on an inter-element level while allowing discontinuities between elements, this flexibility generates a plethora of difficulties when one attempts to post-process DG fields for analysis and evaluation of scientific results. Smoothness-increasing accuracy-conserving (SIAC) filtering enhances the smoothness of the field by eliminating the discontinuity between elements in a way that is consistent with the DG methodology; in particular, high-order accuracy is preserved and in many cases increased. Fundamental to the post-processing approach is the convolution of a spline-based kernel against a DG field. This paper presents a study of the impact of numerical quadrature approximations on the resulting convolution. We discuss both theoretical estimates as well as empirical results which demonstrate the efficacy of the post-processing approach when different levels and types of quadrature approximation are used. Finally, we provide some guidelines for effective use of SIAC filtering of DG fields when used as input to common post-processing and visualization techniques.

Smoothness-increasing accuracy-conserving (SIAC) filters were introduced as a class of postprocessing techniques to ameliorate the quality of numerical solutions of discontinuous Galerkin (DG) simulations. SIAC filtering works to eliminate the oscillations in the error by introducing smoothness back to the DG field and raises the accuracy in the L2 –norm up to its natural superconvergent accuracy in the negative-order norm. The increased smoothness in the filtered DG solutions can then be exploited by simulation postprocessing tools such as streamline integrators where the absence of continuity in the data can lead to erroneous visualizations. However, lack of extension of this filtering technique, both theoretically and computationally, to nontrivial mesh structures along with the expensive core operators have been a hindrance towards the application of the SIAC filters to more realistic simulations. In this dissertation, we focus on the numerical solutions of linear hyperbolic equations solved with the discontinuous Galerkin scheme and provide a thorough analysis of SIAC filtering applied to such solution data. In particular, we investigate how the use of different quadrature techniques could mitigate the extensive processing required when filtering over the whole computational field. Moreover, we provide detailed and efficient algorithms that a numerical practitioner requires to know in order to implement this filtering technique effectively. In our first attempt to expand the application scope of this filtering technique, we demonstrate both mathematically and through numerical examples that it is indeed possible to observe SIAC filtering characteristics when applied to numerical solutions obtained over structured triangular meshes. We further provide a thorough investigation of the interplay between mesh geometry and filtering. Building upon these promising results, we present how SIAC filtering could be applied to gain higher accuracy and smoothness when dealing with totally unstructured triangular meshes. Lastly, we provide the extension of our filtering scheme to structured tetrahedral meshes. Guidelines and future work regarding the application of the SIAC filter in the visualization domain are also presented. We further note that throughout this document, the terms postprocessing and filtering will be used interchangeably.