Multigrid Reduced in Time for Isogeometric Analysis

[EN] Isogeometric Analysis (IgA) can be seen as the natural extension of the Finite Element Method (FEM) to high-order B-spline basis functions. Combined with a time integration scheme within the method of lines, IgA has become a viable alternative to FEM for time-dependent problems. However, as processors’ clock speeds are no longer increasing but the number of cores are going up, traditional (i.e., sequential) time integration schemes become more and more the bottleneck within these large-scale computations. The Multigrid Reduced in Time (MGRIT) method is a parallel-in-time integration method that enables exploitation of parallelism not only in space but also in the temporal direction. In this paper, we apply MGRIT to discretizations arising from IgA for the first time in the literature. In particular, we investigate the (parallel) performance of MGRIT in this context for a variety of geometries, MGRIT hierarchies and time integration schemes. Numerical results show that the MGRIT method converges independent of the mesh width, spline degree of the B-spline basis functions and time step size ∆t and is highly parallelizable when applied in the context of IgA.Tielen, R.; Möller, M.; Vuik, K. (2022). Multigrid Reduced in Time for Isogeometric Analysis. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 47-56. https://doi.org/10.4995/YIC2021.2021.12219OCS475

Reduction of computing time for least-squares migration based on the Helmholtz equation by graphics processing units

In geophysical applications, the interest in leastsquares migration (LSM) as an imaging algorithm is increasing due to the demand for more accurate solutions and the development of high-performance computing. The computational engine of LSM in this work is the numerical solution of the 3D Helmholtz equation in the frequency domain. The Helmholtz solver is Bi-CGSTAB preconditioned with the shifted Laplace matrix-dependent multigrid method. In this paper, an efficient LSM algorithm is presented using several enhancements. First of all, a frequency decimation approach is introduced that makes use of redundant information present in the data. It leads to a speedup of LSM, whereas the impact on accuracy is kept minimal. Secondly, a new matrix storage format Very Compressed Row Storage (VCRS) is presented. It not only reduces the size of the stored matrix by a certain factor but also increases the efficiency of the matrix-vector computations. The effects of lossless and lossy compression with a proper choice of the compression parameters are positive. Thirdly, we accelerate the LSM engine by graphics cards (GPUs). A GPU is used as an accelerator, where the data is partially transferred to a GPU to execute a set of operations or as a replacement, where the complete data is stored in the GPU memory. We demonstrate that using the GPU as a replacement leads to higher speedups and allows us to solve larger problem sizes. Summarizing the effects of each improvement, the resulting speedup can be at least an order of magnitude compared to the original LSM method

Towards Accuracy and Scalability: Combining Isogeometric Analysis with Deflation to Obtain Scalable Convergence for the Helmholtz Equation

Finding fast yet accurate numerical solutions to the Helmholtz equation remains a challenging task. The pollution error (i.e. the discrepancy between the numerical and analytical wave number k) requires the mesh resolution to be kept fine enough to obtain accurate solutions. A recent study showed that the use of Isogeometric Analysis (IgA) for the spatial discretization significantly reduces the pollution error. However, solving the resulting linear systems by means of a direct solver remains computationally expensive when large wave numbers or multiple dimensions are considered. An alternative lies in the use of (preconditioned) Krylov subspace methods. Recently, the use of the exact Complex Shifted Laplacian Preconditioner (CSLP) with a small complex shift has shown to lead to wave number independent convergence while obtaining more accurate numerical solutions using IgA. In this paper, we propose the use of deflation techniques combined with an approximated inverse of the CSLP using a geometric multigrid method. Numerical results obtained for both one- and two-dimensional model problems, including constant and non-constant wave numbers, show scalable convergence with respect to the wave number and approximation order p of the spatial discretization. Furthermore, when kh is kept constant, the proposed approach leads to a significant reduction of the computational time compared to the use of the exact inverse of the CSLP with a small shift

A biomechanical mathematical model for the collagen bundle distribution-dependent contraction and subsequent retraction of healing dermal wounds

A continuum hypothesis-based, biomechanical model is presented for the simulation of the collagen bundle distribution-dependent contraction and subsequent retraction of healing dermal wounds that cover a large surface area. Since wound contraction mainly takes place in the dermal layer of the skin, solely a portion of this layer is included explicitly into the model. This portion of dermal layer is modeled as a heterogeneous, orthotropic continuous solid with bulk mechanical properties that are locally dependent on both the local concentration and the local geometrical arrangement of the collagen bundles. With respect to the dynamic regulation of the geometrical arrangement of the collagen bundles, it is assumed that a portion of the collagen molecules are deposited and reoriented in the direction of movement of (myo)fibroblasts. The remainder of the newly secreted collagen molecules are deposited by ratio in the direction of the present collagen bundles. Simulation results show that the distribution of the collagen bundles influences the evolution over time of both the shape of the wounded area and the degree of overall contraction of the wounded area. Interestingly, these effects are solely a consequence of alterations in the initial overall distribution of the collagen bundles, and not a consequence of alterations in the evolution over time of the different cell densities and concentrations of the modeled constituents. In accordance with experimental observations, simulation results show furthermore that ultimately the majority of the collagen molecules ends up permanently oriented toward the center of the wound and in the plane that runs parallel to the surface of the skin

A mathematical model for the simulation of the contraction of burns

A continuum hypothesis-based model is developed for the simulation of the contraction of burns in order to gain new insights into which elements of the healing response might have a substantial influence on this process. Tissue is modeled as a neo-Hookean solid. Furthermore, (myo)fibroblasts, collagen molecules, and a generic signaling molecule are selected as model components. An overview of the custom-made numerical algorithm is presented. Subsequently, good agreement is demonstrated with respect to variability in the evolution of the surface area of burns over time between the outcomes of computer simulations and measurements obtained in an experimental study. In the model this variability is caused by varying the values for some of its parameters simultaneously. A factorial design combined with a regression analysis are used to quantify the individual contributions of these parameter value variations to the dispersion in the surface area of healing burns. The analysis shows that almost all variability in the surface area can be explained by variability in the value for the myofibroblast apoptosis rate and, to a lesser extent, the value for the collagen molecule secretion rate. This suggests that most of the variability in the evolution of the surface area of burns over time in the experimental study might be attributed to variability in these two rates. Finally, a probabili