The cost of continuity: performance of iterative solvers on isogeometric finite elements

In this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized Galerkin weak form. Specifically, we compare performance of linear solvers when discretizing using $C^0$ B-splines, which span traditional finite element spaces, and $C^{p-1}$ B-splines, which represent maximum continuity. We provide theoretical estimates for the increase in cost of the matrix-vector product as well as for the construction and application of black-box preconditioners. We accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size $h$ and polynomial order of approximation $p$. Finally, we present timing results for a range of preconditioning options for the Laplace problem. We conclude that the matrix-vector product operation is at most \slfrac{33p^2}{8} times more expensive for the more continuous space, although for moderately low $p$, this number is significantly reduced. Moreover, if static condensation is not employed, this number further reduces to at most a value of 8, even for high $p$. Preconditioning options can be up to $p^3$ times more expensive to setup, although this difference significantly decreases for some popular preconditioners such as Incomplete LU factorization

On Stochastic Error and Computational Efficiency of the Markov Chain Monte Carlo Method

In Markov Chain Monte Carlo (MCMC) simulations, the thermal equilibria quantities are estimated by ensemble average over a sample set containing a large number of correlated samples. These samples are selected in accordance with the probability distribution function, known from the partition function of equilibrium state. As the stochastic error of the simulation results is significant, it is desirable to understand the variance of the estimation by ensemble average, which depends on the sample size (i.e., the total number of samples in the set) and the sampling interval (i.e., cycle number between two consecutive samples). Although large sample sizes reduce the variance, they increase the computational cost of the simulation. For a given CPU time, the sample size can be reduced greatly by increasing the sampling interval, while having the corresponding increase in variance be negligible if the original sampling interval is very small. In this work, we report a few general rules that relate the variance with the sample size and the sampling interval. These results are observed and confirmed numerically. These variance rules are derived for the MCMC method but are also valid for the correlated samples obtained using other Monte Carlo methods. The main contribution of this work includes the theoretical proof of these numerical observations and the set of assumptions that lead to them

Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines

Calabr{\`o} et al. [10] changed the paradigm of the mass and stiffness computation from the traditional element-wise assembly to a row-wise concept, showing that the latter one offers integration that may be orders of magnitude faster. Considering a B-spline basis function as a non-negative measure, each mass matrix row is integrated by its own quadrature rule with respect to that measure. Each rule is easy to compute as it leads to a linear system of equations, however, the quadrature rules are of the Newton-Cotes type, that is, they require a number of quadrature points that is equal to the dimension of the spline space. In this work, we propose weighted quadrature rules of Gaussian type which require the minimum number of quadrature points while guaranteeing exactness of integration with respect to the weight function. The weighted Gaussian rules arise as solutions of non-linear systems of equations. We derive rules for the mass and stiffness matrices for uniform $C^1$ quadratic and $C^2$ cubic isogeometric discretizations. In each parameter direction, our rules require locally only $p+1$ quadrature points, $p$ being the polynomial degree. While the nodes cannot be reused for various weight functions as in [10], the computational cost of the mass and stiffness matrix assembly is comparable.RYC-2017-2264

Goal-oriented adaptivity for a conforming residual minimization method in a dual discontinuous Galerkin norm

We propose a goal-oriented mesh-adaptive algorithm for a finite element method stabilized via residual minimization on dual discontinuous-Galerkin norms. By solving a saddle-point problem, this residual minimization delivers a stable continuous approximation to the solution on each mesh instance and a residual projection onto a broken polynomial space, which is a robust error estimator to minimize the discrete energy norm via automatic mesh refinement. In this work, we propose and analyze a goal-oriented adaptive algorithm for this stable residual minimization. We solve the primal and adjoint problems considering the same saddle-point formulation and different right-hand sides. By solving a third stable problem, we obtain two efficient error estimates to guide goal oriented adaptivity. We illustrate the performance of this goal-oriented adaptive strategy on advection-diffusion reaction problems

Parallel refined Isogeometric Analysis in 3D

We study three-dimensional isogeometric analysis (IGA) and the solution of the resulting system of linear equations via a direct solver. IGA uses highly continuous $C^{p-1}$ basis functions, which provide multiple benefits in terms of stability and convergence properties. However, smooth basis significantly deteriorate the direct solver performance and its parallel scalability. As a partial remedy for this, refined Isogeometric Analysis (rIGA) method improves the sequential execution of direct solvers. The refinement strategy enriches traditional highly-continuous $C^{p-1}$ IGA spaces by introducing low-continuity $C^{0}$ 0-hyperplanes along the boundaries of certain pre-defined macro-elements. In this work, propose a solution strategy for rIGA for parallel distributed memory machines and compare the computational costs of solving rIGA vs IGA discretizations. We verify our estimates with parallel numerical experiments. Results show that the weak parallel scalability of the direct solver improves approximately by a factor of $p^{2}$ when considering rIGA discretizations rather than highly-continuous IGA spaces

Numerical simulation of spheres moving and colliding close to bed streams, with a complete characterization of turbulence

River morphodynamics and sediment transportMechanics of sediment transpor

Reliability of Mainstream Tablets for 2D Analysis of a Drop Jump Landing

Please refer to the pdf version of the abstract located adjacent to the title

Multi-level analysis of on-chip optical wireless links

Networks-on-chip are being regarded as a promising solution to meet the on-going requirement for higher and higher computation capacity. In view of future kilo-cores architectures, electrical wired connections are likely to become inefficient and alternative technologies are being widely investigated. Wireless communications on chip may be therefore leveraged to overcome the bottleneck of physical interconnections. This work deals with wireless networks-on-chip at optical frequencies, which can simplify the network layout and reduce the communication latency, easing the antenna on-chip integration process at the same time. On the other end, optical wireless communication on-chip can be limited by the heavy propagation losses and the possible cross-link interference. Assessment of the optical wireless network in terms of bit error probability and maximum communication range is here investigated through a multi-level approach. Manifold aspects, concurring to the final system performance, are simultaneously taken into account, like the antenna radiation properties, the data-rate of the core-to core communication, the geometrical and electromagnetic layout of the chip and the noise and interference level. Simulations results suggest that communication up to some hundreds of μm can be pursued provided that the antenna design and/or the target data-rate are carefully tailored to the actual layout of the chip