Solving time‐dependent PDEs using the material point method, a case study from gas dynamics

Tran, L.T.; Kim, J.; Berzins, Martin

doi:10.1002/fld.2031

Cited by 22 publications

(18 citation statements)

References 24 publications

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“…Here, we continue to expand on the analysis in [22] to help to understand the relationship between decreasing t and the expected behavior of the differencex n+1 p − x n+1 p in (76). To simplify, the second term (in the square brackets) is merely the projection of grid acceleration onto the particle at time n: a n p .…”

Section: Discussionmentioning

confidence: 96%

“…The difference between the two-step and one-step particle positions,x k+1 p − x k+1 p , or the timestepping jump error, was calculated in [22]. They showed this difference to be:…”

Section: Discussionmentioning

confidence: 99%

“…In particular, the acceleration calculated using the MPM algorithm may have significant quadrature errors in space and discontinuities in time [22], both of which make second-order temporal convergence unrealizable to the MPM practitioner. Figure 2 shows a sample of a typical grid acceleration field a(x) = i i (x)a i encountered in standard MPM when piecewise-linear basis functions are used.…”

Section: Interpreting the Coupling Of Lagrangian And Eulerian Simulatmentioning

confidence: 97%

“…The recent work by Tran et al [22] analyzed errors in an MPM algorithm with respect to a gas dynamics problem. One feature of their MPM implementation which differs from most other implementations is a volume normalization step.…”

Section: Impact Of Spatial Discontinuities On Time-steppingmentioning

confidence: 99%

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Decoupling and balancing of space and time errors in the material point method (MPM)

Steffen

Kirby

Berzins

2009

Numerical Meth Engineering

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SUMMARYThe material point method (MPM) is a computationally effective particle method with mathematical roots in both particle-in-cell and finite element-type methods. The method has proven to be extremely useful in solving solid mechanics problems involving large deformations and/or fragmentation of structures, problem domains that are sometimes problematic for finite element-type methods. Recently, the MPM community has focused significant attention on understanding the basic mathematical error properties of the method.Complementary to this thrust, in this paper we show how spatial and temporal errors are typically coupled within the MPM framework. In an attempt to overcome the challenge to analysis that this coupling poses, we take advantage of MPM's connection to finite element methods by developing a 'moving-mesh' variant of MPM that allows us to use finite element-type error analysis to demonstrate and understand the spatial and temporal error behaviors of MPM. We then provide an analysis and demonstration of various spatial and temporal errors in MPM and in simplified MPM-type simulations.Our analysis allows us to anticipate the global error behavior in MPM-type methods and allows us to estimate the time-step where spatial and temporal errors are balanced. Larger time-steps result in solutions dominated by temporal errors and show second-order temporal error convergence. Smaller time-steps result in solutions dominated by spatial errors, and hence temporal refinement produces no appreciative change in the solution. Based upon our understanding of MPM from both analysis and numerical experimentation, we are able to provide to MPM practitioners a collection of guidelines to be used in the selection of simulation parameters that respect the interplay between spatial (grid) resolution, number of particles and time-step.

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Section: Discussionmentioning

confidence: 96%

“…The difference between the two-step and one-step particle positions,x k+1 p − x k+1 p , or the timestepping jump error, was calculated in [22]. They showed this difference to be:…”

Section: Discussionmentioning

confidence: 99%

Section: Interpreting the Coupling Of Lagrangian And Eulerian Simulatmentioning

confidence: 97%

Section: Impact Of Spatial Discontinuities On Time-steppingmentioning

confidence: 99%

See 2 more Smart Citations

Decoupling and balancing of space and time errors in the material point method (MPM)

Steffen

Kirby

Berzins

2009

Numerical Meth Engineering

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“…When there are very high property gradients, such as at shocks, numerical oscillation tends to propagate and diminish accuracy; therefore, it requires the development of an effective algorithm to control it. Tran et al (2010) utilised an artificial diffusion term to reduce numerical noise and oscillations. Problems were seen with local extrema in velocity, so a 'smoothing' scheme was utilised where values at a point become dependent upon the adjacent values, effectively averaging out the value at the centre point.…”

mentioning

confidence: 99%

An effective limiting algorithm for particle-based numerical simulations of compressible flows

Mason

Chen

et al. 2011

International Journal of Computational Fluid Dynamics

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Eulerian computational fluid dynamics (CFD) and Lagrangian computational structural dynamics (CSD) are used extensively in the aerospace industry. Combined mesh-based Eulerian and particle-based Lagrangian algorithms are very effective for modelling and simulation due to the increased efficiency of combining the two numerical simulations. However, when compressible flows are simulated using a particle-based algorithm, calculations of strong discontinuity, such as a shock wave, may become unstable. In the present study, a numerical limiter is integrated with a particle-based CFD code to remedy this instability. The limiting algorithm incorporates an 'averaging' technique which calculates average values using the properties of neighbouring particles (also known as material points), including mass, momentum and energy. These averaged values are then input to a min-mode limiter to eliminate numerical noise and incur dissipation in the flow in areas with steep property gradients. The results of this algorithm show very stable solutions with minimal oscillations when applied to the one-dimensional shock tube problem and an increased accuracy with reduced oscillations for a two-dimensional cylinder cross-flow problem.

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Scalable large‐scale fluid–structure interaction solvers in the Uintah framework via hybrid task‐based parallelism algorithms

Meng

Berzins

2013

Concurrency and Computation

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Uintah is a software framework that provides an environment for solving fluid-structure interaction problems on structured adaptive grids for 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 a novel asynchronous task-based approach with fully 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. The Uintah approach to the growth of the number of core counts per socket together with the prospect of less memory per core is to adopt a model that uses MPI to communicate between nodes and a shared memory model on-node so as to achieve scalability on large-scale systems. For this approach to be successful, it is necessary to design data structures that large numbers of cores can simultaneously access without contention. 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 260K cores. ‡ Titan is a parallel computer under construction at Oak Ridge National Laboratory with about 299K CPU cores now with a large number of attached GPUs to be added in 2012. § Stampede is a parallel computer under construction at Texas Advanced Computing Center with two petaflops of CPU performance and eight petaflops of accelerator performance. 1389This work will start to address some of these challenges by developing scheduling algorithms and associated software that will be used to tackle fluid-structure interaction algorithms in which mesh refinement is used to resolve the structure. The vehicle for this work is the Uintah Computational Framework [2][3][4]. The broad range and challenging nature of Uintah applications and the software itself are described in [5]. The Uintah code was designed to solve fluid-structure interaction problems arising from a number of physical scenarios. Uintah uses a combination of computational fluid dynamics (CFD) algorithms in its implicit continuous-fluid Eulerian (ICE) solver [6,7] and couples ICE to a particle-based solver for solids known as the material point method (MPM) [8,9]. The adaptive mesh refinement (AMR) approach used in Uintah is that of multiple levels of regularly refined mesh patches. This mesh is also used as a scratch pad for the MPM calculation of the movement and deformation of the solid [10,11]. Uintah has been shown to scale successfully with many fluid and solid problems with adaptive meshes [12,13]; however, the scalability of fluid-structure interaction problems has proven to be somewhat more challenging. This is at least partly because the part...

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Solving time‐dependent PDEs using the material point method, a case study from gas dynamics

Cited by 22 publications

References 24 publications

Decoupling and balancing of space and time errors in the material point method (MPM)

Decoupling and balancing of space and time errors in the material point method (MPM)

An effective limiting algorithm for particle-based numerical simulations of compressible flows

Scalable large‐scale fluid–structure interaction solvers in the Uintah framework via hybrid task‐based parallelism algorithms

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