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We present the Unified Form Language (UFL), which is a domain-specific language for representing weak formulations of partial differential equations with a view to numerical approximation. Features of UFL include support for variational forms and functionals, automatic differentiation of forms and expressions, arbitrary function space hierarchies for multifield problems, general differential operators and flexible tensor algebra. With these features, UFL has been used to effortlessly express finite element methods for complex systems of partial differential equations in near-mathematical notation, resulting in compact, intuitive and readable programs. We present in this work the language and its construction. An implementation of UFL is freely available as an open-source software library. The library generates abstract syntax tree representations of variational problems, which are used by other software libraries to generate concrete low-level implementations. Some application examples are presented and libraries that support UFL are highlighted.

Background and Purpose-Wall shear stress (WSS) and pressure are important factors in the development of cerebral aneurysms. We aimed to develop a computational fluid dynamics simulator for flow in the complete circle of Willis to study the impact of variations in vessel radii and bifurcation angles on WSS and pressure on vessel walls. Methods-Blood flow was modeled with Navier-Stokes equations as an incompressible newtonian fluid within rigid vessel walls. A model of the circle of Willis geometry was approximated as a network of tubes around cubic curves. Pulsatile inlet flow rates and constant outlet pressure were used as boundary conditions. Results-The simulations confirmed that differences in vessel radii and asymmetric branch angles influence WSS magnitude and spatial distribution. High WSS occurred at locations where aneurysms are frequent and in anatomic variants known to be associated with an increased risk for aneurysm development. Conclusions-Computational fluid dynamics analysis can be applied to the complete circle of Willis and should be used to study the pathophysiology of this complex vascular structure, including risk factors for aneurysm development. Key Words: aneurysm Ⅲ computational fluid dynamics Ⅲ circle of Willis Ⅲ hemodynamics Ⅲ wall shear stress D isruption of the internal elastic lamina is required for the creation of saccular aneurysms. Hemodynamic factors play an important role in this process. Saccular aneurysms usually arise at the distal carina of bifurcations, where vessels are exposed to the maximum impact of wall shear stress (WSS). 1 The amount of WSS depends on the geometry of the bifurcation. 2-5 WSS is minimized when the relation between vessel radii and bifurcation angles follows optimality principles of minimum work. 6 -8 In the circle of Willis, there is a confluence of flow from 3 vessels: both internal carotid arteries and the basilar artery (BA). Therefore, the hemodynamics in the circle of Willis is anatomically significantly different from the hemodynamics in normal branching situations addressed by the optimality principle. Accordingly, the normal physiology of flow and the likely impact of deviation from normality in the circle of Willis are not fully understood.In a previous study, we analyzed 3-dimensional digital subtraction angiography images of cerebral vessels with respect to vessel radii and bifurcation angles and concluded that bifurcations beyond the circle of Willis approximated optimality principles, whereas those within the circle of Willis did not. 9 In addition, we observed an increased prevalence of aneurysms at bifurcations with large branch angles. Furthermore, studies of this complex vascular structure in patients, animal models, or experimental in vitro models are difficult. Therefore, simulations with computational fluid dynamics (CFD) may contribute to the understanding of this problem. In the present study, we aimed to develop a CFD simulator for flow in the complete circle of Willis to study the impact of variations in vessel radii and bifu...

Several cardiovascular diseases are caused from localised abnormal blood flow such as in the case of stenosis or aneurysms. Prevailing theories propose that the development is caused by abnormal wall shear stress in focused areas. Computational fluid mechanics have arisen as a promising tool for a more precise and quantitative analysis, in particular because the anatomy is often readily available even by standard imaging techniques such as magnetic resonance and computed tomography angiography. However, computational fluid mechanics rely on accurate initial and boundary conditions, which are difficult to obtain. In this paper, we address the problem of recovering high-resolution information from noisy and low-resolution physical measurements of blood flow (for example, from phase-contrast magnetic resonance imaging [PC-MRI]) using variational data assimilation based on a transient Navier-Stokes model. Numerical experiments are performed in both 3D (2D space and time) and 4D (3D space and time) and with pulsatile flow relevant for physiological flow in cerebral aneurysms. The results demonstrate that, with suitable regularisation, the model accurately reconstructs flow, even in the presence of significant noise.

Over the last fifty years, the finite element method has emerged as a successful methodology for solving a wide range of partial differential equations. At the heart of any finite element simulation is the assembly of matrices and vectors from finite element variational forms. In this paper, we present a general and unified framework for finite element assembly. Based on this framework, we propose a specific software interface called Unified Form-assembly Code (UFC) between problem-specific and general-purpose components of finite element programs. The interface is general in the sense that it applies to a wide range of finite element problems (including mixed finite elements and discontinuous Galerkin methods) and may be used with libraries that differ widely in their design. The interface consists of a minimal set of abstract C++ classes and data transfer is via plain C arrays.We discuss how one may use the UFC interface to build a plug-and-play system for finite element simulation where basic components such as computational meshes, linear algebra and, in particular, variational form evaluation may come from different libraries and be used interchangeably. We further discuss how the UFC interface is used to glue together components from the FEniCS suite of software to provide an integrated highlevel environment where variational forms may be entered as expressions directly in Python and assembled efficiently into sparse matrices.A central design goal for the interface is to minimize dependency on external libraries for the problem-specific code used in applications. Thus, the UFC interface consists of a single C++ header file and does not rely on external libraries for its operation. In particular, the UFC interface does not depend on any other FEniCS components. As a result, finite element code developers may use the interface to detach equationspecific details from general-purpose library code, allowing very flexible connections to alternative libraries. We encourage developers of finite element libraries to incorporate the interface in their libraries. The UFC interface is released into the public domain.