Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits

We analyze stability and convergence of sequential implicit methods for coupled flow and geomechanics, in which the flow problem is solved first. We employ the von Neumann and energy methods for linear and nonlinear problems, respectively. We consider two sequential methods with the generalized midpoint rule for t_n+α, where α is the parameter of time discretization: namely, the fixed-strain and fixed-stress splits. The von Neumann method indicates that the fixed-strain split is only conditionally stable, and that its stability limit is a coupling strength less than unity if α ? 0.5. On the other hand, the fixed-stress split is unconditionally stable when α ? 0.5, the amplification factors of the fixed-stress split are different from those of the undrained split and are identical to the fully coupled method. Unconditional stability of the fixed-stress split is also obtained from the energy method for poroelastoplasticity. We show that the fixed-stress split is contractive and B-stable when α ? 0.5.We also estimate the convergence behaviors for the two sequential methods by the matrix based and spectral analyses for the backward Euler method in time. From the estimates, the fixed-strain split may not be convergent with a fixed number of iterations particularly around the stability limit even though it is stable. The fixed-stress split, however, is convergent for a fixed number of iterations, showing better accuracy than the undrained split. Even when we cannot obtain the exact local bulk modulus (or exact rock compressibility) at the flow step a priori due to complex boundary conditions or the nonlinearity of the materials, the fixed-stress split can still provide stability and convergence by an appropriate estimation of the local bulk modulus, such as the dimension-based estimation, by which the employed local bulk modulus is less stiff than the exact local bulk modulus.We provide numerical examples supporting all the estimates of stability and convergence for the fixed-strain and fixed-stress splits.     

Computing Local Artin Maps,and Solvability of Norm Equations

Let L = K(α) be an Abelian extension of degree n of a number field K, given by the minimal polynomial of α over K. We describe an algorithm for computing the local Artin map associated with the extension L / K at a finite or infinite prime v of K. We apply this algorithm to decide if a nonzero a ∈ K is a norm from L, assuming that L / K is cyclic.     

A new class of massively parallel direction splitting for the incompressible Navier–Stokes equations

We introduce in this paper a new direction splitting algorithm for solving the incompressible Navier–Stokes equations. The main originality of the method consists of using the operator (I ? ?_xx)(I ? ?_yy)(I ? ?_zz) for approximating the pressure correction instead of the Poisson operator as done in all the contemporary projection methods. The complexity of the proposed algorithm is significantly lower than that of projection methods, and it is shown the have the same stability properties as the Poisson-based pressure-correction techniques, either in standard or rotational form. The first-order (in time) version of the method is proved to have the same convergence properties as the classical first-order projection techniques. Numerical tests reveal that the second-order version of the method has the same convergence rate as its second-order projection counterpart as well. The method is suitable for parallel implementation and preliminary tests show excellent parallel performance on a distributed memory cluster of up to 1024 processors. The method has been validated on the three-dimensional lid-driven cavity flow using grids composed of up to 2 × 10⁹ points.     

Finite-state dimension and real arithmetic

We use entropy rates and Schur concavity to prove that, for every integer k ⩾ 2, every nonzero rational number q, and every real number α, the base-k expansions of α, q + α, and qα all have the same finite-state dimension and the same finite-state strong dimension. This extends, and gives a new proof of, Wall’s 1949 theorem stating that the sum or product of a nonzero rational number and a Borel normal number is always Borel normal.     

A Hermite cubic collocation scheme for plane strain hydraulic fractures

We describe a novel cubic Hermite collocation scheme for the solution of the coupled integro-partial differential equations governing the propagation of a hydraulic fracture in a state of plane strain. Special blended cubic Hermite-power–law basis functions, with arbitrary index 0 < α < 1, are developed to treat the singular behavior of the solution that typically occurs at the tips of a hydraulic fracture. The implementation of blended infinite elements to model semi-infinite crack problems is also described. Explicit formulae for the integrated kernels associated with the cubic Hermite and blended basis functions are provided. The cubic Hermite collocation algorithm is used to solve a number of different test problems with two distinct propagation regimes and the results are shown to converge to published similarity and asymptotic solutions. The convergence rate of the cubic Hermite scheme is determined by the order of accuracy of the tip asymptotic expansion as well as the O(h⁴) error due to the Hermite cubic interpolation. The errors due to these two approximations need to be matched in order to achieve optimal convergence. Backward Euler time-stepping yields a robust algorithm that, along with geometric increments in the time-step, can be used to explore the transition between propagation regimes over many orders of magnitude in time.     

Hybrid real-coded genetic algorithm for data partitioning in multi-round load distribution and scheduling in heterogeneous systems

Data partitioning and scheduling is one the important issues in minimizing the processing time for parallel and distributed computing system. We consider a single-level tree architecture of the system and the case of affine communication model, for a general m processor system with n rounds of load distribution. For this case, there exists an optimal activation order, optimal number of processors m* (m * ≤ m), and optimal rounds of load distribution n* (n * ≤ n), such that the processing time of the entire processing load is a minimum. This is a difficult optimization problem because for a given activation order, we have to first identify the processors that are participating (in the computation process) in every round of load distribution and then obtain the load fractions assigned to them, and the processing time. Hence, in this paper, we propose a real-coded genetic algorithm (RCGA) to solve the optimal activation order, optimal number of processors m* (m * ≤ m), and optimal rounds of load distribution n* (n * ≤ n), such that the processing time of the entire processing load is a minimum. RCGA employs a modified crossover and mutation operators such that the operators always produce a valid solution. Also, we propose different population initialization schemes to improve the convergence. Finally, we present a comparative study with simple real-coded genetic algorithm and particle swarm optimization to highlight the advantage of the proposed algorithm. The results clearly indicate the effectiveness of the proposed real-coded genetic algorithm.     

A finite element method with discontinuous rotations for the Mindlin–Reissner plate model

We present a continuous-discontinuous finite element method for the Mindlin–Reissner plate model based on continuous polynomials of degree k ? 2 for the transverse displacements and discontinuous polynomials of degree k ? 1 for the rotations. We prove a priori convergence estimates, uniformly in the thickness of the plate, and thus show that locking is avoided. We also derive a posteriori error estimates based on duality, together with corresponding adaptive procedures for controlling linear functionals of the error. Finally, we present some numerical results.     

Modal analysis and modified cascade neural networks in identification of geometrical parameters of circular arches

A hybrid computational system, composed of the finite element method (FEM) and cascade neural network system (CNNs), is applied to the identification of three geometrical parameters of elastic arches, i.e. span l, height f and cross-sectional thickness h. FEM is used in the direct (forward) analysis, which corresponds to the mapping α = {l, f, h} → {ω_j}, where: α – vector of control parameters, ω_j – arch eigenfrequencies. The reverse analysis is related to the identification procedure in which the reverse mapping is performed {ω_j} → {α_i}. For the identification purposes a recurrent, three level CNNs of structure (D^k-H^k-1)_s was formulated, where: k – recurrence step, s = I, II, III-levels of cascade system. The Semi-Bayesian approach is introduced for the design of CNNs applying the MML Maximum Marginal Likelihood) criterion. The computation of hyperparameters is performed by means of the Bayesian procedure evidence. The numerical analysis proves a great numerical efficiency of the proposed hybrid approach for both the perfect (noiseless) values of eigenfrequencies and noisy ones simulated by an added artificial noise.     

Fuzzy logic and statistical-based modelling of the Marshall Stability of asphalt concrete under varying temperatures and exposure times

In this study, the Marshall Stability (MS) of asphalt concrete under varying temperature and exposure times was modelled by using fuzzy logic and statistical method. This is an experimental study conducted using statistics and fuzzy logic methods. In order to investigate the Marshall Stability of asphalt concrete based on exposure time and environment temperature, exposure times of 1.5, 3, 4.5 and 6 h and temperatures of 30, 40 and 50 °C were selected. The MS of the asphalt concrete at 17 °C (in laboratory environment temperature) was used as reference. The results showed that the MS of the asphalt core samples decreased 40.16% at 30 °C after 1.5 h and 62.39% after 6 h. At 40 °C the decrease was 74.31% after 1.5 h, and 78.10% after 6 h. At 50 °C the stability of the asphalt decreased 83.22% after 1.5 h, 88.66% after 6 h. The relationships between experimental results, fuzzy logic model and statistical results exhibited good correlation. The correlation coefficient was R = 0.99 for fuzzy logic model and R² = 0.9 for statistical method. Based on the results of the study, it could be said that both the fuzzy logic method and statistical methods could be used for modelling of the stability of asphalt concrete under varying temperature and exposure time.     

Generalized Strong Pseudoprime Tests and Applications

We describe probabilistic primality tests applicable to integers whose prime factors are all congruent to 1 mod r where r is a positive integer;r =  2 is the Miller–Rabin test. We show that if ν rounds of our test do not find n ≠  =  (r +  1)²composite, then n is prime with probability of error less than (2 r) ^− ν. Applications are given, first to provide a probabilistic primality test applicable to all integers, and second, to give a test for values of cyclotomic polynomials.     

Jetmax: Scalable max–min congestion control for high-speed heterogeneous networks

Recent surge of interest towards congestion control that relies on single-link feedback (e.g., XCP, RCP, MaxNet, EMKC, VCP), suggests that such systems may offer certain benefits over traditional models of additive packet loss. Besides topology-independent stability and faster convergence to efficiency/fairness, it was recently shown that any stable single-link system with a symmetric Jacobian tolerates arbitrary fixed, as well as time-varying, feedback delays. Although delay-independence is an appealing characteristic, the EMKC system developed in exhibits undesirable equilibrium properties and slow convergence behavior. To overcome these drawbacks, we propose a new method called JetMax and show that it admits a low-overhead implementation inside routers (three additions per packet), overshoot-free transient and steady state, tunable link utilization, and delay-insensitive flow dynamics. The proposed framework also provides capacity-independent convergence time, where fairness and utilization are reached in the same number of RTT steps for a link of any bandwidth. Given a 1 mb/s, 10 gb/s, or googol (10¹⁰⁰) bps link, the method converges to within 1% of the stationary state in six RTTs. We finish the paper by comparing JetMax’s performance to that of existing methods in ns2 simulations and discussing its Linux implementation.     

Bounds for the Roots of Lacunary Polynomials

A polynomial P(X)  = X^d + a_d − 1X^d − 1 + ⋯ is called lacunary when a_d − 1 =  0. We give bounds for the roots of such polynomials with complex coefficients. These bounds are much smaller than for general polynomials.     

Computational assessment of several hydrogen-free high energy compounds

Tetrazino-tetrazine-tetraoxide (TTTO) is an attractive high energy compound, but unfortunately, it is not yet experimentally synthesized so far. Isomerization of TTTO leads to its five isomers, bond-separation energies were empolyed to compare the global stability of six compounds, it is found that isomer 1 has the highest bond-separation energy (1204.6 kJ/mol), compared with TTTO (1151.2 kJ/mol); thermodynamic properties of six compounds were theoretically calculated, including standard formation enthalpies (solid and gaseous), standard fusion enthalpies, standard vaporation enthalpies, standard sublimation enthalpies, lattice energies and normal melting points, normal boiling points; their detonation performances were also computed, including detonation heat (Q, cal/g), detonation velocity (D, km/s), detonation pressure (P, GPa) and impact sensitivity (h₅₀, cm), compared with TTTO (Q = 1311.01 J/g, D = 9.228 km/s, P = 40.556 GPa, h₅₀ = 12.7 cm), isomer 5 exhibites better detonation performances (Q = 1523.74 J/g, D = 9.389 km/s, P = 41.329 GPa, h₅₀ =  28.4 cm).     

Irregular Primes and Cyclotomic Invariants to 12 Million

Computations of irregular primes and associated cyclotomic invariants were extended to all primes up to 12 million using multisectioning/convolution methods and a novel approach which originated in the study of Stickelberger codes Shokrollahi (1996). The latter idea reduces the problem to that of finding zeros of a polynomial overF_pof degree  <  (p −  1) / 2 among the quadratic nonresidues mod p. Use of fast polynomial gcd-algorithms gives anO (p log²p loglog p)-algorithm for this task. A more efficient algorithm, with comparable asymptotic running time, can be obtained by using Schönhage–Strassen integer multiplication techniques and fast multiple polynomial evaluation algorithms; this approach is particularly efficient when run on primes p for whichp −  1 has small prime factors. We also give some improvements on previous implementations for verifying the Kummer–Vandiver conjecture and for computing the cyclotomic invariants of a prime.     

Inclusion complexation of pinostrobin with various cyclodextrin derivatives

Pinostrobin (PNS) is one of the important flavonoids and can be abundantly found in the rhizomes of fingerroot (Boesenbergia rotrunda) and galangal (Alpinia galangal and Alpinia officinarum), the herbal basis of Southeast Asian cooking. Similar to other flavonoids, PNS exhibits anti-oxidative, anti-inflammatory and anti-cancer properties. However, this compound has an extremely low water solubility that limits its use in pharmaceutical applications. Beta-cyclodextrin (βCD) and its derivatives, 2,6-dimethyl-βCD (2,6-DMβCD) and the three hydroxypropyl-βCDs (2-HPβCD, 6-HPβCD and 2,6-DHPβCD), have unique properties that enhance the stability and solubility of such low-soluble guest molecules. In the present study, molecular dynamics simulations were applied to investigate the dynamics and stability of PNS inclusion complexes with βCD and its derivatives (2,6-DMβCD, 2,6-DHPβCD, 2-HPβCD and 6-HPβCD). PNS was able to form complexes with βCD and all four of its derivatives by either the chromone (C-PNS) or phenyl (P-PNS) ring dipping toward the cavity. According to the molecular mechanics-generalized Born surface area binding free energy values, the stability of the different PNS/βCD complexes was ranked as 2,6-DHPβCD > 2,6-DMβCD > 2-HPβCD > 6-HPβCD > βCD. These theoretical results were in good agreement with the stability constants that had been determined by the solubility method.     

Deadline-based scheduling in support of real-time data delivery

The use of deadline-based scheduling in support of real-time delivery of application data units (ADUs) in a packet-switched network is investigated. Of interest is priority scheduling where a packet with a smaller ratio of T/H (time until delivery deadline over number of hops remaining) is given a higher priority. We refer to this scheduling algorithm as the T/H algorithm. T/H has time complexity of O(log N) for a backlog of N packets and was shown to achieve good performance in terms of the percentage of ADUs that are delivered on-time. We develop a new and efficient algorithm, called T/H − p, that has O(1) time complexity. The performance difference of T/H, T/H − p and FCFS are evaluated by simulation. Implementations of T/H and T/H − p in high-speed routers are also discussed. We show through simulation that T/H − p is superior to FCFS but not as good as T/H. In view of the constant time complexity, T/H − p is a good candidate for high-speed routers when both performance and implementation cost are taken into consideration.     

On reliability of the folded hypercubes

In this paper, we explore the 2-extra connectivity and 2-extra-edge-connectivity of the folded hypercube FQ_n. We show that κ₂(FQ_n) = 3n − 2 for n ⩾ 8; and λ₂(FQ_n) = 3n − 1 for n ⩾ 5. That is, for n ⩾ 8 (resp. n ⩾ 5), at least 3n − 2 vertices (resp. 3n − 1 edges) of FQ_n are removed to get a disconnected graph that contains no isolated vertices (resp. edges). When the folded hypercube is used to model the topological structure of a large-scale parallel processing system, these results can provide more accurate measurements for reliability and fault tolerance of the system.     

Differences in computer exposure between university administrators and CAD draftsmen

This study utilized an external logger system for onsite measurements of computer activities of two professional groups—twelve university administrators and twelve computer-aided design (CAD) draftsmen. Computer use of each participant was recorded for 10 consecutive days—an average of 7.9 ± 1.8 workdays and 7.8 ± 1.5 workdays for administrators and draftsmen, respectively. Quantitative parameters computed using recorded data were daily dynamic duration (DD) and static duration, daily keystrokes, mouse clicks, wheel scrolling counts, mouse movement and dragged distance, average typing and clicking rates, and average time holding down keys and mouse buttons. Significant group differences existed in the number of daily keystrokes (p < 0.0005) and mouse clicks (p < 0.0005), mouse distance moved (p < 0.0005), typing rate (p < 0.0001), daily mouse DD (p < 0.0001), and keyboard DD (p < 0.005). Both groups had significantly longer mouse DD than keyboard DD (p < 0.0001). Statistical analysis indicates that the duration of computer use for different computer tasks cannot be represented by a single formula with same set of quantitative parameters as those associated with mouse and keyboard activities. Results of this study demonstrate that computer exposure during different tasks cannot be estimated solely by computer use duration. Quantification of onsite computer activities is necessary when determining computer-associated risk of musculoskeletal disorders. Other significant findings are discussed.     

A ranking procedure based on a natural monotonicity constraint

We formulate a new ranking procedure in the traditional context where each voter has expressed a linear order relation or ranking over the candidates. The final ranking of the candidates is taken to be the one which best adheres to a natural monotonicity constraint. For a ranking a ≻ b ≻ c, monotonicity implies that the strength with which a ≻ c is supported should not be less than the strength with which either one of a ≻ b or b ≻ c is supported. We investigate some properties of this ranking procedure and encounter some surprising preliminary results.