Implemented Solver Research Articles

Implementation and Validation of a Numerical Method for Concentrated Suspensions in Large Flows Based on the Particle Diffusion Equation

This work presents the development of an OpenFOAM solver aimed at correctly predicting dynamics of concentrated suspensions when subjected to non-uniform shear flows. The newly implemented solver is able to predict the behavior of a heterogeneous mixture whose characteristics depend on the solid particle local concentration. To simulate such behavior, the conservation equation expressing the time variation of the particle volume fraction has been implemented in OpenFOAM; this was achieved by modifying a pre-existing solver, pimpleFoam, which discretizes the Navier–Stokes system of equation through the PIMPLE algorithm. As a first step, the formulation of the momentum equation has been adapted to correctly solve cases with non-Newtonian fluids. Successively, the Krieger’s correlation has been used to model the viscosity variation in the domain to take in account heterogeneous particle distributions. Finally, the iterative cycle for the solution of the migration equation has been included within the time loop. The above-mentioned code has been successfully validated by comparing the numerical results with the measured data provided by experiments reported in literature.

Local Volume Solvers for Earth System Data Assimilation: Implementation in the Framework for Joint Effort for Data Assimilation Integration

AbstractThe Joint Effort for Data assimilation Integration (JEDI) is an international collaboration aimed at developing an open software ecosystem for model agnostic data assimilation. This paper considers implementation of the model‐agnostic family of the local volume solvers in the JEDI framework. The implemented solvers include the Local Ensemble Transform Kalman Filter (LETKF), the Gain form of the Ensemble Transform Kalman Filter (GETKF), and the optimal interpolation variant of the LETKF (LETKF‐OI). This paper documents the implementation strategy for the family of the local volume solvers within the JEDI framework. We also document an expansive set of localization approaches that includes generic distance‐based localization, localization based on modulated ensemble products, and localizations specific to ocean (based on the Rossby radius of deformation), and land (based on the terrain difference between observation and model grid point). Finally, we apply the developed solvers in a limited set of experiments, including single‐observation experiments in atmosphere and ocean, and cycling experiments for the atmosphere, ocean, land, and aerosol assimilation. We also illustrate how JEDI Ensemble Kalman Filter solvers can be used in a strongly coupled framework using the interface solver approximation, which provides increments to the ocean based on observations from the ocean and atmosphere.

Automated translation and accelerated solving of differential equations on multiple GPU platforms

We demonstrate a high-performance vendor-agnostic method for massively parallel solving of ensembles of ordinary differential equations (ODEs) and stochastic differential equations (SDEs) on GPUs. The method is integrated with a widely used differential equation solver library in a high-level language (Julia’s DifferentialEquations.jl) and enables GPU acceleration without requiring code changes by the user. Our approach achieves state-of-the-art performance compared to hand-optimized CUDA-C++ kernels while performing 20–100× faster than the vectorizing map (vmap) approach implemented in JAX and PyTorch. Performance evaluation on NVIDIA, AMD, Intel, and Apple GPUs demonstrates performance portability and vendor-agnosticism. We show composability with MPI to enable distributed multi-GPU workflows. The implemented solvers are fully featured – supporting event handling, automatic differentiation, and incorporation of datasets via the GPU’s texture memory – allowing scientists to take advantage of GPU acceleration on all major current architectures without changing their model code and without loss of performance. We distribute the software as an open-source library, DiffEqGPU.jl.

A Level Set-Based Solver for Two-Phase Incompressible Flows: Extension to Magnetic Fluids

Development of a two-phase incompressible solver for magnetic flows in the magnetostatic case is presented. The proposed numerical toolkit couples the Navier–Stokes equations of hydrodynamics with Maxwell's equations of electromagnetism to model the behaviour of magnetic flows in the presence of a magnetic field. To this end, a rigorous implementation of a second-order two-phase solver for incompressible nonmagnetic flows is introduced first. This solver is implemented in the finite-difference framework, where a fifth-order conservative level set method is employed to capture the evolution of the interface, along with an incompressible solver based on the projection scheme to model the fluids. The solver demonstrates excellent performance even with high density ratios across the interface (Atwood number ≈ 1 ), while effectively preserving the mass conservation property. Subsequently, the numerical discretisation of Maxwell's equations under the magnetostatic assumption is described in detail, utilising the vector potential formulation. The primary second-order solver for two-phase flows is extended to the case of magnetic flows, by incorporating the Lorentz force into the momentum equation, accounting for high magnetic permeability ratios across the interface. The implemented solver is then utilised for examining the deformation of ferrofluid droplets in both quiescent and shear flow regimes across various susceptibility values of the droplets. The results suggest that increasing the susceptibility value of the ferrofluid droplet can affect its deformation and rotation in low capillary regimes. In higher capillary flows, increasing the magnetic permeability jump across the interface can further lead to droplet breakup as well. The effect of this property is also investigated for the Rayleigh–Taylor instability growth in magnetic fluids.

Implementation of a velocity decomposition method coupled with volume-of-fluid method for simulating free-surface flows

An efficient velocity decomposition method (VDM) is proposed for simulating free surface flows based on OpenFOAM platform, which is primarily intended for marine hydrodynamics applications. In this method, it is assumed that the free surface can be treated as a source region, which introduces rotating velocity related to the wave elevation. In the implemented solver called cpVDFoam, the decomposed potential flow velocity is computed by solving the Laplace equation without considering the effect of free surface, and the remaining non-potential velocity is determined with a complimentary Navier–Stokes (N-S) equation. Further, the free surface is modeled using volume-of-fluid (VOF) method in the viscous domain for obtaining the non-potential velocity. Flow field analysis shows that non-potential velocity magnitude contributed by freesurface is positive correlation with wave elevation, and which value shall decrease to zero in the zero wave elevation region. The efficacy of the proposed VDM is validated by comparing the simulation results with the results obtained using standard OpenFOAM solver interFoam and earlier reported experiment. All results indicate that the present method exhibited similar accuracy as the conventional CFD method but less sensitive to the domain size.

High-Order CFD Solvers on Three-Dimensional Unstructured Meshes: Parallel Implementation of RKDG Method with WENO Limiter and Momentum Sources

In aerospace engineering, high-order computational fluid dynamics (CFD) solvers suitable for three-dimensional unstructured meshes are less developed than expected. The Runge–Kutta discontinuous Galerkin (RKDG) finite element method with compact weighted essentially non-oscillatory (WENO) limiters is one of the candidates, which might give high-order solutions on unstructured meshes. In this article, we provide an efficient parallel implementation of this method for simulating inviscid compressible flows. The implemented solvers are tested on lower-dimensional model problems and real three-dimensional engineering problems. Results of lower-dimensional problems validate the correctness and accuracy of these solvers. The capability of capturing complex flow structures even on coarse meshes is shown in the results of three-dimensional applications. For solving problems containing rotary wings, an unsteady momentum source model is incorporated into the solvers. Such a finite element/momentum source hybrid method eliminates the reliance on advanced mesh techniques, which might provide an efficient tool for studying rotorcraft aerodynamics.

Databionic Swarm Intelligence to Screen Wastewater Recycling Quality with Factorial and Hyper-Parameter Non-Linear Orthogonal Mini-Datasets

Electrodialysis (ED) may be designed to enhance wastewater recycling efficiency for crop irrigation in areas where water distribution is otherwise inaccessible. ED process controls are difficult to manage because the ED cells need to be custom-built to meet local requirements, and the wastewater influx often has heterogeneous ionic properties. Besides the underlying complex chemical phenomena, recycling screening is a challenge to engineering because the number of experimental trials must be maintained low in order to be timely and cost-effective. A new data-centric approach is presented that screens three water quality indices against four ED-process-controlling factors for a wastewater recycling application in agricultural development. The implemented unsupervised solver must: (1) be fine-tuned for optimal deployment and (2) screen the ED trials for effect potency. The databionic swarm intelligence classifier is employed to cluster the L9(34) OA mini-dataset of: (1) the removed Na+ content, (2) the sodium adsorption ratio (SAR) and (3) the soluble Na+ percentage. From an information viewpoint, the proviso for the factor profiler is that it should be apt to detect strength and curvature effects against not-computable uncertainty. The strength hierarchy was analyzed for the four ED-process-controlling factors: (1) the dilute flow, (2) the cathode flow, (3) the anode flow and (4) the voltage rate. The new approach matches two sequences for similarities, according to: (1) the classified cluster identification string and (2) the pre-defined OA factorial setting string. Internal cluster validity is checked by the Dunn and Davies–Bouldin Indices, after completing a hyper-parameter L8(4122) OA screening. The three selected hyper-parameters (distance measure, structure type and position type) created negligible variability. The dilute flow was found to regulate the overall ED-based separation performance. The results agree with other recent statistical/algorithmic studies through external validation. In conclusion, statistical/algorithmic freeware (R-packages) may be effective in resolving quality multi-indexed screening tasks of intricate non-linear mini-OA-datasets.

PicFoam: An OpenFOAM based electrostatic Particle-in-Cell solver

picFoam is a fully kinetic electrostatic Particle-in-Cell (PIC) solver, including Monte Carlo Collisions (MCC), for non-equilibrium plasma research in the open-source framework of OpenFOAM. The solver’s modular design, based on the same principles used in OpenFOAM, makes it highly flexible, by allowing the user to choose different methods at run time, and extendable, by building upon templated modular classes. The implementation of the PIC method employing the finite volume method, allows it to simulate on arbitrary geometries in one to three dimensions. OpenFOAM’s barycentric particle tracking is used effectively to perform charge and field weighting from the Lagrangian particle based description to the Eulerian field description and backwards without computational expensive particle searching algorithm. picFoam also includes open and general circuit boundary models for the description of real plasma devices. Program summaryProgram Title: picFoamCPC Library link to program files:http://dx.doi.org/10.17632/bbsm8tjgjy.1Developer’s repository link:https://github.com/TFDzarm/picFoamLicensing provisions: GPLv3Programming language: C++Nature of problem: The description of equilibrium and non-equilibrium plasma with its complex collective behavior between charged species. As well as their interaction with circuits connected to the boundaries of the plasma. The applications are versatile and lie e.g. in the description of plasma phenomena like discharges in arcjet electric propulsion systems, as well as a general study of plasma phenomena.Solution method: Implementation of the Particle-in-Cell method with Monte Carlo Collisions [1] in the open-source numerical toolbox OpenFOAM [2]. By this the newly implemented solver gains access to OpenFOAM’s powerful tool sets and an easy to set up simulation case structure. Fields are solved by employing OpenFOAM’s cell-centered finite volume solvers, this allows for the simulation on arbitrary structured meshes. The leapfrog scheme [1] is employed for the integration of the particle’s equation of motion combined with relativistic and non-relativistic integration schemes for Newton’s second law of motion [3]. Monte Carlo Collision methods include models for the collision of electrons and neutral species, considered collision events are elastic collision events and inelastic excitation and ionization events. Additional models include Coulomb collisions and isotropic scattering of neutral species [4]. The solver is able to bind circuit models, include open circuit, ideal voltage and current sources, as well as general purpose circuits with an ideal voltage source and a resistance, an impedance and a capacity in series, to the plasma domain [5].Additional comments including restrictions and unusual features: The finite volume method is a relatively rarely used method for implementations of the PIC method, here by employing this method we gain the ability to use arbitrary structured meshes in one to three dimensions. OpenFOAM’s barycentric particle tracking is a novel approach for moving the particles through the mesh. With this method we are able to interpolate (weight) the charges efficiently to the mesh without using computational expensive algorithms to locate the particle’s positions beforehand.

The ghost fluid method for OpenFOAM: A comparative study in marine context

The open source CFD library OpenFOAM is utilized to implement the ghost fluid method (GFM). The implemented solver GFMFoam is compared against the standard volume-of-fluid (VOF) solver interFoam in OpenFOAM. For both VOF solvers, the scalar transport equation is solved similarly but the discretization of the momentum and continuity equations differ between GFMFoam and interFoam. The two methods are compared in four test cases with mild surface deformation. First, the GFM implementation is validated. Second, the results on a 2 d interface shear layer highlight the importance of the viscous term treatment in the GFM. Third, the model scale Wiggley hull results indicate that the GFMFoam is more sensitive to the grid aspect ratio and resolution than interFoam but the viscous resistance is better predicted. Also, GFMFoam is shown to be less sensitive to numerical ventilation than interFoam. Fourth, the results from the Hamburg Test Case indicate a total resistance prediction within 1% when evaluated against reference data at model scale and 10% at full scale. Based on the steady state test cases of the present study, GFMFoam has moderate benefits over interFoam but it does not clearly outperform interFoam.

Scheduling Solution for Space Debris Observations

The scheduling model of orbital debris can be considered as composed by a set of optical ground station which move with the surface of the Earth, a set of space situation awareness centre which can be assumed connected to ground station, and orbital debris travelling through different kind of orbit generating visibility windows when the line of sight (LOS) to ground station exists. To solve this, scheduling problem can be considered as the allocation of observing time of the telescope to collect measurements of the debris (i.e., to improve orbit or attitude determination, spin rate evolution, etc.). The increasing number of space debris to be observed with optical ground station has caused the combinatorial explosion in the number of intervals to be scheduled. Therefore, new scheduling approach to provide a solution to the new requests is needed. In the framework of the Agreement between Italian Space Agency (ASI) and National Institute of Astrophysics (INAF) “Supporto alle attivita IADC e validazione pre-operativa per SST” (N.2015-028-R.0), S5Lab research group has developed a network of observatory fully dedicated for space debris observations. Moreover, a scheduler called NICO (Networked Instrument Coordinator for debris Observations) has been developed specifically designed for the harmonization of individual user requests by considering meteorological and astronomical constraints. This paper outlines the advantages of the developed custom solution based on modular architecture. NICO’s architecture is based on a multilayer system: the front end is designed to allow external registered users to specify their observing requests and to assign specific scientific priority; the back-end is changed for the business logic by determining the visibility windows for each request and to solve the conflict. The implemented conflict solver solution is based on genetic algorithms. This paper outlines the results of the application of NICO scheduler to the 2018-IADC (Inter-Agency Space Debris Coordination Committee) international optical observing campaigns.

Implementation of an Implicit Solver in ADCIRC Storm Surge Model

The current state of science does not offer any remedy to stop a hurricane from occurring. Therefore, accurate storm surge models capable of predicting water velocity and elevation are indispensable. In this paper, the implementation of an implicit solver in the Advanced Circulation (ADCIRC) storm surge model is presented. The implemented implicit solver uses hybrid finite element and finite volume techniques for solving shallow water equations. Objectives of this research include: Enhancing numerical stability, providing an option of using large timesteps, and the usage of a relatively easier mathematical formulation than the existing one in ADCIRC. The storm surge hindcast of Hurricane Katrina that hit Louisiana and Mississippi in 2005 is used as a case study. Stability of the solver, comparison of water elevation and velocity against observed data, impact of timestep sizes, and execution times of solvers are thoroughly investigated in this study. Results of the implemented implicit solver are compared with those of existing lumped explicit and semi-implicit solvers of ADCIRC; the findings appear to be very promising.

Implementing Efficient All Solutions SAT Solvers

All solutions SAT (AllSAT for short) is a variant of the propositional satisfiability problem. AllSAT has been relatively unexplored compared to other variants despite its significance. We thus survey and discuss major techniques of AllSAT solvers. We accurately implemented them and conducted comprehensive experiments using a large number of instances and various types of solvers including a few publicly available software. The experiments revealed the solvers’ characteristics. We made our implemented solvers publicly available so that other researchers can easily develop their solvers by modifying our code and comparing it with existing methods.

Improvement of a subviral particle tracker by the use of a LAP-Kalman-algorithm

AbstractAutomated detection and tracking of subviral particles is a promising method to obtain insights in complicated virus-cell interactions. This paper describes the implementation of a linear assignment problem solver and a Kalman-filter in an existing particle tracking algorithm. Two different simulated image sequences are used for the evaluation of the algorithms. Tracking and detection results of the new implemented solver are compared to the results of the original algorithm. The improved algorithm is able to improve the results by closing gaps in the particle tracks.

Structural design of high-performance capacitive accelerometers using parametric optimization with uncertainties

ABSTRACTElectrostatic or capacitive accelerometers are among the highest volume microelectromechanical systems (MEMS) products nowadays. The design of such devices is a complex task, since they depend on many performance requirements, which are often conflicting. Therefore, optimization techniques are often used in the design stage of these MEMS devices. Because of problems with reliability, the technology of MEMS is not yet well established. Thus, in this work, size optimization is combined with the reliability-based design optimization (RBDO) method to improve the performance of accelerometers. To account for uncertainties in the dimensions and material properties of these devices, the first order reliability method is applied to calculate the probabilities involved in the RBDO formulation. Practical examples of bulk-type capacitive accelerometer designs are presented and discussed to evaluate the potential of the implemented RBDO solver.

Towards Next Generation Sequential and Parallel SAT Solvers

This thesis focuses on improving the SAT solving technology. The improvements focus on two major subjects: sequential SAT solving and parallel SAT solving. To better understand sequential SAT algorithms, the abstract reduction system Generic CDCL is introduced. With Generic CDCL, the soundness of solving techniques can be modeled. Next, the conflict driven clause learning algorithm is extended with the three techniques local look-ahead, local probing and all UIP learning that allow more global reasoning during search. These techniques improve the performance of the sequential SAT solver Riss. Then, the formula simplification techniques bounded variable addition, covered literal elimination and an advanced cardinality constraint extraction are introduced. By using these techniques, the reasoning of the overall SAT solving tool chain becomes stronger than plain resolution. When using these three techniques in the formula simplification tool Coprocessor before using Riss to solve a formula, the performance can be improved further. Due to the increasing number of cores in CPUs, the scalable parallel SAT solving approach iterative partitioning has been implemented in Pcasso for the multi-core architecture. Related work on parallel SAT solving has been studied to extract main ideas that can improve Pcasso. Besides parallel formula simplification with bounded variable elimination, the major extension is the extended clause sharing level based clause tagging, which builds the basis for conflict driven node killing. The latter allows to better identify unsatisfiable search space partitions. Another improvement is to combine scattering and look-ahead as a superior search space partitioning function. In combination with Coprocessor, the introduced extensions increase the performance of the parallel solver Pcasso. The implemented system turns out to be scalable for the multi-core architecture. Hence iterative partitioning is interesting for future parallel SAT solvers. Constraints (2015) 20:504–505 DOI 10.1007/s10601-015-9226-6 * Norbert Manthey norbert.manthey@tu-dresden.de 1 TU Dresden, Dresden, Germany The implemented solvers participated in international SAT competitions. In 2013 and 2014 Pcasso showed a good performance. Riss in combination with Coprocessor won several first, second and third prices, including two Kurt-Godel-Medals. Hence, the introduced algorithms improved modern SAT solving technology. School: TU Dresden Supervisors: Prof. Steffen Holldober Graduated: Monday, December 1, 2014 Link to full text: http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-158672 Notes: Nominated for ACP Doctoral Research Award, and GI-Dissertationspreis 2014 Constraints (2015) 20:504–505 505

Towards next generation sequential and parallel SAT solvers

This thesis focuses on improving the SAT solving technology. The improvements focus on two major subjects: sequential SAT solving and parallel SAT solving. To better understand sequential SAT algorithms, the abstract reduction system Generic CDCL is introduced. With Generic CDCL, the soundness of solving techniques can be modeled. Next, the conflict driven clause learning algorithm is extended with the three techniques local look-ahead, local probing and all UIP learning that allow more global reasoning during search. These techniques improve the performance of the sequential SAT solver Riss. Then, the formula simplification techniques bounded variable addition, covered literal elimination and an advanced cardinality constraint extraction are introduced. By using these techniques, the reasoning of the overall SAT solving tool chain becomes stronger than plain resolution. When using these three techniques in the formula simplification tool Coprocessor before using Riss to solve a formula, the performance can be improved further. Due to the increasing number of cores in CPUs, the scalable parallel SAT solving approach iterative partitioning has been implemented in Pcasso for the multi-core architecture. Related work on parallel SAT solving has been studied to extract main ideas that can improve Pcasso. Besides parallel formula simplification with bounded variable elimination, the major extension is the extended clause sharing level based clause tagging, which builds the basis for conflict driven node killing. The latter allows to better identify unsatisfiable search space partitions. Another improvement is to combine scattering and look-ahead as a superior search space partitioning function. In combination with Coprocessor, the introduced extensions increase the performance of the parallel solver Pcasso. The implemented system turns out to be scalable for the multi-core architecture. Hence iterative partitioning is interesting for future parallel SAT solvers. Constraints (2015) 20:504–505 DOI 10.1007/s10601-015-9226-6 * Norbert Manthey norbert.manthey@tu-dresden.de 1 TU Dresden, Dresden, Germany The implemented solvers participated in international SAT competitions. In 2013 and 2014 Pcasso showed a good performance. Riss in combination with Coprocessor won several first, second and third prices, including two Kurt-Godel-Medals. Hence, the introduced algorithms improved modern SAT solving technology. School: TU Dresden Supervisors: Prof. Steffen Holldober Graduated: Monday, December 1, 2014 Link to full text: http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-158672 Notes: Nominated for ACP Doctoral Research Award, and GI-Dissertationspreis 2014 Constraints (2015) 20:504–505 505

Multi-GPU Accelerated Admittance Method for High-Resolution Human Exposure Evaluation.

A multi-graphics processing unit (GPU) accelerated admittance method solver is presented for solving the induced electric field in high-resolution anatomical models of human body when exposed to external low-frequency magnetic fields. In the solver, the anatomical model is discretized as a three-dimensional network of admittances. The conjugate orthogonal conjugate gradient (COCG) iterative algorithm is employed to take advantage of the symmetric property of the complex-valued linear system of equations. Compared against the widely used biconjugate gradient stabilized method, the COCG algorithm can reduce the solving time by 3.5 times and reduce the storage requirement by about 40%. The iterative algorithm is then accelerated further by using multiple NVIDIA GPUs. The computations and data transfers between GPUs are overlapped in time by using asynchronous concurrent execution design. The communication overhead is well hidden so that the acceleration is nearly linear with the number of GPU cards. Numerical examples show that our GPU implementation running on four NVIDIA Tesla K20c cards can reach 90 times faster than the CPU implementation running on eight CPU cores (two Intel Xeon E5-2603 processors). The implemented solver is able to solve large dimensional problems efficiently. A whole adult body discretized in 1-mm resolution can be solved in just several minutes. The high efficiency achieved makes it practical to investigate human exposure involving a large number of cases with a high resolution that meets the requirements of international dosimetry guidelines.

An SMT-based approach to weak controllability for disjunctive temporal problems with uncertainty

The framework of temporal problems with uncertainty (TPU) is useful to express temporal constraints over a set of activities subject to uncertain (and uncontrollable) duration. In this work, we focus on the most general class of TPU, namely disjunctive TPU (DTPU), and consider the case of weak controllability, that allows one to model problems arising in practical scenarios (e.g. on-line scheduling).We first tackle the decision problem, i.e. whether there exists a schedule of the activities that, depending on the uncertainty, satisfies all the constraints. We propose a logical approach, based on the reduction to a problem of Satisfiability Modulo Theories (SMT), in the theory of Linear Real Arithmetic with Quantifiers. This results in the first implemented solver for weak controllability of DTPUs.Then, we tackle the problem of synthesizing control strategies for scheduling the activities. We focus on strategies that are amenable for efficient execution. We prove that linear strategies are not always sufficient, even in the sub-case of simple TPU (STPU), while piecewise-linear strategies, that are multiple conditionally-applied linear strategies, are always sufficient. We present several algorithms for the synthesis of linear and piecewise-linear strategies, in case of STPU and of DTPU.All the algorithms are implemented on top of SMT solvers. We provide experimental evidence of the scalability of the proposed techniques, with dramatic speed-ups in strategy execution compared to on-line reasoning.

On finite volume method implementation of poro‐elasto‐plasticity soil model

SummaryAccurate prediction of the interactions between the nonlinear soil skeleton and the pore fluid under loading plays a vital role in many geotechnical applications. It is therefore important to develop a numerical method that can effectively capture this nonlinear soil‐pore fluid coupling effect. This paper presents the implementation of a new finite volume method code of poro‐elasto‐plasticity soil model. The model is formulated on the basis of Biot's consolidation theory and combined with a perfect plasticity Mohr‐Coulomb constitutive relation. The governing equation system is discretized in a segregated manner, namely, those conventional linear and uncoupled terms are treated implicitly, while those nonlinear and coupled terms are treated explicitly by using any available values from previous time or iteration step. The implicit–explicit discretization leads to a linearized and decoupled algebraic system, which is solved using the fixed‐point iteration method. Upon the convergence of the iterative method, fully nonlinear coupled solutions are obtained. Also explored in this paper is the special way of treating traction boundary in finite volume method compared with FEM. Finally, three numerical test cases are simulated to verify the implementation procedure. It is shown in the simulation results that the implemented solver is capable of and efficient at predicting reasonable soil responses with pore pressure coupling under different loading situations. Copyright © 2015 John Wiley & Sons, Ltd.

Adaptive step ODE algorithms for the 3D simulation of electric heart activity with graphics processing units

In this paper we studied the implementation and performance of adaptive step methods for large systems of ordinary differential equations systems in graphics processing units, focusing on the simulation of three-dimensional electric cardiac activity. The Rush–Larsen method was applied in all the implemented solvers to improve efficiency. We compared the adaptive methods with the fixed step methods, and we found that the fixed step methods can be faster while the adaptive step methods are better in terms of accuracy and robustness.