Time integration errors and energy conservation properties of the Stormer Verlet method applied to MPM

The success of the Material Point Method (MPM) in solving many challenging problems nevertheless raises some open questions regarding the fundamental properties of the methodsuchastheenergyconservationsincebeingaddressedbyBardenhagenandbyLoveand Sulsky. Similarly while low order symplectic time integration techniques are used with MPM, higher order methods have not been used. For this reason the Stormer Verlet method, a popular and widely-used symplectic method is applied to MPM. Both the time integration error and the energy conservation properties of this method applied to MPM are considered. The method is showntohavelocallythirdorderaccuracyofenergyconservationintime. Thisisincontrastto the locally second order accuracy in energy conservation of the methods that are used in many MPM calculations. This third accuracy accuracy is demonstrated both locally and globally on a standard MPM test example

Nonlinear stability of the mpm method

The Material Point Method (MPM) has been very successful in providing solutions to many challenging problems involving large deformations. The nonlinear nature of MPM makes it necessary to use a full nonlinear stability analysis to determine a stable timestep. The stability analysis of Spigler and Vianello is adapted to MPM and used to derive a stable timestep bound for a model problem. This bound is contrasted against a traditional CFL bound

Data and range-bounded polynomials in ENO methods

journal articleEssentially Non-Oscillatory (ENO) methods and Weighted Essentially Non- Oscillatory (WENO) methods are of fundamental importance in the numerical solution of hyperbolic equations. A key property of such equations is that the solution must remain positive or lie between bounds. A modification of the polynomials used in ENO methods to ensure that the modified polynomials are either bounded by adjacent values (data-bounded) or lie within a specified range (range-bounded) is considered. It is shown that this approach helps both in the range boundedness in the preservation of extrema in the ENO polynomial solution

Uintah hybrid task-based parallelism algorithm

pre-printAbstract-Uintah is a software framework that provides an environment for solving large-scale science and engineering problems involving the solution of partial differential equations. Uintah uses a combination of fluid-flow solvers and particle-based methods for solids, together with adaptive meshing and asynchronous task-based approach with automated load balancing. When applying Uintah to fluid-structure interaction problems, the combination of adaptive meshing and the movement of structures through space present a formidable challenge in terms of achieving scalability on large-scale parallel computers. Adopting a model that uses MPI to communicate between nodes and a shared memory model on-node is one approach to achieve scalability on large-scale systems.This scalability challenge is addressed here for Uintah, by the development of new hybrid runtime and scheduling algorithms combined with novel lock-free data structures, making it possible for Uintah to achieve excellent scalability for a challenging fluid-structure problem with mesh refinement on as many as 256K cores

A survey of high level frameworks in block-structured adaptive mesh refinement packages

pre-printOver the last decade block-structured adaptive mesh refinement (SAMR) has found increasing use in large, publicly available codes and frameworks. SAMR frameworks have evolved along different paths. Some have stayed focused on specific domain areas, others have pursued a more general functionality, providing the building blocks for a larger variety of applications. In this survey paper we examine a representative set of SAMR packages and SAMR-based codes that have been in existence for half a decade or more, have a reasonably sized and active user base outside of their home institutions, and are publicly available. The set consists of a mix of SAMR packages and application codes that cover a broad range of scientific domains. We look at their high-level frameworks, their design trade-offs and their approach to dealing with the advent of radical changes in hardware architecture. The codes included in this survey are BoxLib, Cactus, Chombo, Enzo, FLASH, and Uintah

Adaptive high-order discontinuous galerkin solution of elastohydrodynamic lubrication point contact problems

ManuscriptThis paper describes an adaptive implementation of a high order Discontinuous Galerkin (DG) method for the solution of elastohydrodynamic lubrication (EHL) point contact problems. These problems arise when modelling the thin lubricating film between contacts which are under sufficiently high pressure that the elastic deformation of the contacting elements cannot be neglected. The governing equations are highly nonlinear and include a second order partial differential equation that is derived via the thin-film approximation. Furthermore, the problem features a free boundary, which models where cavitation occurs, and this is automatically captured as part of the solution process. The need for spatial adaptivity stems from the highly variable length scales that are present in typical solutions. Results are presented which demonstrate both the effectiveness and the limitations of the proposed adaptive algorithm

DAG-based software frameworks for PDEs

pre-printThe task-based approach to software and parallelism is well-known and has been proposed as a potential candidate, named the silver model, for exas-cale software. This approach is not yet widely used in the large-scale multi-core parallel computing of complex systems of partial differential equations. After surveying task-based approaches we investigate how well the Uintah software and an extension named Wasatch fit in the task-based paradigm and how well they perform on large scale parallel computers. The conclusion is that these approaches show great promise for petascale but that considerable algorithmic challenges remain

Systematic debugging methods for large-scale HPC computational frameworks

pre-printParallel computational frameworks for high-performance computing are central to the advancement of simulation-based studies in science and engineering. Finding and fixing bugs in these frameworks can be time consuming. If left unchecked, these bugs diminish the amount of new science performed. A systematic study of the Uintah Computational Framework investigates debugging approaches, leveraging the framework's modular structure

Investigating applications portability with the Uintah DAG-based runtime system on PetaScale supercomputers

pre-printPresent trends in high performance computing present formidable challenges for applications code using multicore nodes possibly with accelerators and/or co-processors and reduced memory while still attaining scalability. Software frameworks that execute machine-independent applications code using a runtime system that shields users from architectural complexities offer a possible solution. The Uintah framework for example, solves a broad class of large-scale problems on structured adaptive grids using fluid-flow solvers coupled with particle-based solids methods. Uintah executes directed acyclic graphs of computational tasks with a scalable asynchronous and dynamic runtime system for CPU cores and/or accelerators/coprocessors on a node. Uintah's clear separation between application and runtime code has led to scalability increases of 1000x without significant changes to application code. This methodology is tested on three leading Top500 machines; OLCF Titan, TACC Stampede and ALCF Mira using three diverse and challenging applications problems. This investigation of scalability with regard to the different processors and communications performance leads to the overall conclusion that the adaptive DAG-based approach provides a very powerful abstraction for solving challenging multi-scale multi-physics engineering problems on some of the largest and most powerful computers available today