Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces

We develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation.

On Numerical Methods for Second-Order Nonlinear Ordinary Differential Equations (ODEs): A Reduction To A System Of First-Order ODEs

2nd-order ODEs can be found in many applications, e.g., motion of pendulum, vibrating springs, etc. We first convert the 2nd-order nonlinear ODEs to a system of 1st-order ODEs which is easier to deal with. Then, Adams-Bashforth (AB) methods are used to solve the resulting system of 1st-order ODE. AB methods are chosen since they are the explicit schemes and more efficient in terms of shorter computational time. However, the step size is more restrictive since these methods are conditionally stable. We find two test cases (one test problem and one manufactured solution) to be used to validate the AB methods. The exact solution for both test cases are available for the error and convergence analysis later on. The implementation of 1st-, 2nd- and 3rd-order AB methods are done using Octave. The error was computed to retrieve the order of convergence numerically and the CPU time was recorded to analyze their efficiency.

A Comparison between Two Dilute Citrate Solutions (15 vs. 18 mmol/l) in Continuous Renal Replacement Therapy: The Base Excess and Renal Substitution Solution Study

Background/Aims: The study aimed to compare the changes in biochemistry occurring in patients undergoing continuous renal replacement therapy (CRRT) using 2 trisodium citrate solutions, Baxter hemofiltration fluid containing 18 mmol/l (C18) and Baxter NamSol, a custom manufactured solution containing 15 mmol/l (C15), both delivered as regional citrate anticoagulation (RCA) predilution fluids for hemofiltration. Methods: This is a prospective randomized control trial conducted in a major regional adult intensive care unit. Patients were randomized to 1 of 2 RCA fluids. Progress was monitored using a standard daily panel of acid-base and biochemical tests. Results: Forty-eight patients, 23 C18 and 25 C15, were recruited. In both groups, acidosis resolved within 36 h of institution of CRRT. By day 3, there were significant differences in serum [Na+], standard base excess and serum bicarbonate concentration, all being higher in the C18 group (p < 0.01). By day 5, the PaCO2 had also risen in the C18 group (p = 0.03). Conclusions: The C15 solution provided equivalent filter life to the C18 solution but without significant hypernatremia and metabolic alkalosis.

The Least Squares Spectral Element Method for the Navier-Stokes and Cahn-Hilliard Equations

The least squares spectral element method (LS-SEM) offers many advantages in the implementation of the finite element model compared with the traditional weak Galerkin method. In this article, the LS-SEM is used to solve the Navier-Stokes (NS) and the Cahn-Hilliard (CH) equations. The NS equation is solved with both C0 and C1 basis functions and their performance is compared in terms of accuracy. A two-dimensional steady-state solver is verified with the case of Kovasznay flow and validated for the cavity flow, and a two-dimensional unsteady solver is verified by a transient manufactured solution case. The phenomenon of phase separation in binary system is described by the CH equation. Due to the fourth-order characteristics of the CH equation, only a high order continuity approximation is used by employing C1 basis function for both space and time domain. The obtained solutions are in accordance with previous results from the literature and show the fundamental characteristics of the NS and CH equations. The results in this study give the possibility of developing a solver for the coupled NS and CH equations.

Demonstration of Emulator-Based Bayesian Calibration of Safety Analysis Codes: Theory and Formulation

System codes for simulation of safety performance of nuclear plants may contain parameters whose values are not known very accurately. New information from tests or operating experience is incorporated into safety codes by a process known as calibration, which reduces uncertainty in the output of the code and thereby improves its support for decision-making. The work reported here implements several improvements on classic calibration techniques afforded by modern analysis techniques. The key innovation has come from development of code surrogate model (or code emulator) construction and prediction algorithms. Use of a fast emulator makes the calibration processes used here with Markov Chain Monte Carlo (MCMC) sampling feasible. This work uses Gaussian Process (GP) based emulators, which have been used previously to emulate computer codes in the nuclear field. The present work describes the formulation of an emulator that incorporates GPs into a factor analysis-type or pattern recognition-type model. This “function factorization” Gaussian Process (FFGP) model allows overcoming limitations present in standard GP emulators, thereby improving both accuracy and speed of the emulator-based calibration process. Calibration of a friction-factor example using a Method of Manufactured Solution is performed to illustrate key properties of the FFGP based process.

Code Verification of ReFRESCO With a Statistically Periodic Manufactured Solution

This paper presents a Code Verification study performed with the unsteady ensemble-averaged Navier-Stokes (URANS) solver ReFRESCO using the Method of Manufactured Solutions. The study uses a statistically periodic manufactured solution including the un-damped eddy-viscosity of the Spalart & Allmaras turbulence model. Three main aspects of the numerical calculations of unsteady flows are addressed in this study: iterative errors; discretization errors (space and time) and the determination of the observed order of (space and time) convergence. The availability of an exact solution allows the determination of the numerical error and so the effects of iterative and discretization errors can be addressed. The paper presents grid and time refinement studies with different (iterative) convergence criteria and demonstrates that grid and time resolution are strongly connected when attempts are made to minimize the numerical uncertainty in the calculation of unsteady flows. The paper also addresses error estimation based on power series expansions in the calculation of unsteady (space and time dependent) flows. Simultaneous grid and time refinement is compared to grid refinement with fixed time step and time refinement with fixed grid. The advantages and limitations of both options are discussed in the context of Code Verification (error evaluation) and Solution Verification (error estimation).

Manufactured solutions and the verification of three-dimensional Stokes ice-sheet models

The manufactured solution technique is used for the verification of computational models in many fields. In this paper, we construct manufactured solutions for the three-dimensional, isothermal, nonlinear Stokes model for flows in glaciers and ice sheets. The solution construction procedure starts with kinematic boundary conditions and is mainly based on the solution of a first-order partial differential equation for the ice velocity that satisfies the incompressibility condition. The manufactured solutions depend on the geometry of the ice sheet, basal sliding parameters, and ice softness. Initial conditions are taken from the periodic geometry of a standard problem of the ISMIP-HOM benchmark tests. The upper surface is altered through the manufactured solution procedure to generate an analytic solution for the time-dependent flow problem. We then use this manufactured solution to verify a parallel, high-order accurate, finite element Stokes ice-sheet model. Simulation results from the computational model show good convergence to the manufactured analytic solution.