Models of cardiac mechanics are increasingly used to investigate cardiac physiology. These models are characterized by a high level of complexity, including the particular anisotropic material properties of biological tissue and the actively contracting material. A large number of independent simulation codes have been developed, but a consistent way of verifying the accuracy and replicability of simulations is lacking. To aid in the verification of current and future cardiac mechanics solvers, this study provides three benchmark problems for cardiac mechanics. These benchmark problems test the ability to accurately simulate pressure-type forces that depend on the deformed objects geometry, anisotropic and spatially varying material properties similar to those seen in the left ventricle and active contractile forces. The benchmark was solved by 11 different groups to generate consensus solutions, with typical differences in higher-resolution solutions at approximately 0.5%, and consistent results between linear, quadratic and cubic finite elements as well as different approaches to simulating incompressible materials. Online tools and solutions are made available to allow these tests to be effectively used in verification of future cardiac mechanics software.
Flow in fractured porous media occurs in the earth's subsurface, in biological tissues, and in man-made materials. Fractures have a dominating influence on flow processes, and the last decade has seen an extensive development of models and numerical methods that explicitly account for their presence. To support these developments, four benchmark cases for single-phase flow in three-dimensional fractured porous media are presented. The cases are specifically designed to test the methods' capabilities in handling various complexities common to the geometrical structures of fracture networks. Based on an open call for participation, results obtained with 17 numerical methods were collected. This paper presents the underlying mathematical model, an overview of the features of the participating numerical methods, and their performance in solving the benchmark cases.
Understanding seismic attenuation and modulus dispersion mechanisms in fractured rocks can result in significant advances for the indirect characterization of such environments. In this paper, we study attenuation and modulus dispersion of seismic waves caused by fluid pressure diffusion (FPD) in stochastic 2‐D fracture networks, allowing for a state‐of‐the‐art representation of natural fracture networks by a power law length distribution. To this end, we apply numerical upscaling experiments consisting of compression and shear tests to our samples of fractured rocks. The resulting P and S wave attenuation and modulus dispersion behavior is analyzed with respect to the density, the length distribution, and the connectivity of the fractures. We focus our analysis on two manifestations of FPD arising in fractured rocks, namely, fracture‐to‐background FPD at lower frequencies and fracture‐to‐fracture FPD at higher frequencies. Our results indicate that FPD is sensitive not only to the fracture density but also to the geometrical characteristics of the fracture length distributions. In particular, our study suggests that information about the local connectivity of a fracture network could be retrieved from seismic data. Conversely, information about the global connectivity, which is directly linked to the effective hydraulic conductivity of the probed volume, remains rather difficult to infer.
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