Pieter D Boom - Profile on Academia.edu (original) (raw)

Papers by Pieter D Boom

This paper investigates the application of discrete exterior calculus (DEC) to predict the materi... more This paper investigates the application of discrete exterior calculus (DEC) to predict the material performance of concrete (mortar and aggregates). The aim is to simulate the discrete and heterogenous structure of concrete directly to better predict local phenomena and their impact on apparent global properties. Towards this goal, an existing DEC formulation of linear elasticity is extended to describe incremental elastic-plastic material behavior with isotropic strain hardening. A Voronoi tessellation of the physical domain is used to represent different constituents of the concrete, where each cell is assigned a local material model and material properties. The interaction of the cells is described using the Delaunay dual tetrahedralization of the tessellation. Constructing the mesh in this order required a new boundary closure for the DEC formulation, which is also presented herein. The formulation is validated through simulation and compared to finite element analysis obtained from Abaqus. Simulations include compression of cubical specimens composed of 1) mortar with uniform properties and elastic-plastic response; 2) mortar, as before, but with different volume fractions of aggregate added, having purely elastic response; as well as 3) some initial simulations with the formation of cracks from specimens in tension. Excellent results are obtained when compared to the finite-element analysis, laying the foundation to simulate more complex phenomena in the future.

Computer Physics Communications, Oct 1, 2022

A formulation of elliptic boundary value problems is used to develop the first discrete exterior ... more A formulation of elliptic boundary value problems is used to develop the first discrete exterior calculus (DEC) library for massively parallel computations with 3D domains. This can be used for steady-state analysis of any physical process driven by the gradient of a scalar quantity, e.g. temperature, concentration, pressure or electric potential, and is easily extendable to transient analysis. In addition to offering this library to the community, we demonstrate one important benefit from the DEC formulation: effortless introduction of strong heterogeneities and discontinuities. These are typical for real materials, but challenging for widely used domain discretization schemes, such as finite elements. Specifically, we demonstrate the efficiency of the method for calculating the evolution of thermal conductivity of a solid with a growing crack population. Future development of the library will deal with transient problems, and more importantly with processes driven by gradients of vector quantities.

Microstructures, physical processes, and discrete differential forms

Procedia Structural Integrity

SoftwareX

New high-performance computing (HPC) software designed for massively parallel computers with high... more New high-performance computing (HPC) software designed for massively parallel computers with high-speed interconnects is presented to accelerate research into geometric formulations of solid mechanics based on discrete exterior calculus (DEC). DEC is a relatively new and entirely discrete approach being developed to model non-smooth material processes, for which continuum descriptions fail. Until now, progress has been slowed by limited HPC software. The tool presented herein integrates the DEC library ParaGEMS into the well-established parallel finite-element (FE) code ParaFEM, leveraging ParaFEM's diverse IO routines, optimised solvers, and interfaces to third-party libraries. This is accomplished by interpreting FE elements, or their subdivision, as independent DEC simplicial complexes. The element-wise contribution to the global system matrix is then replaced with the DEC formalism, superimposing contributions from the dual mesh at element boundaries. The integrated tool is validated using five miniApps for scalar diffusion and linear elasticity on synthetic microstructures with emerging discontinuities, showing the performance for both continuum and discrete problems. Profiling indicates DEC calculations have excellent scaling and the solver achieves approximately 80% parallel efficiency using naïve partitioning on ∼8000 cores with >135 million unknowns. The tool is now being used to develop DEC formulations of more complex phenomena, such as material nonlinearity and fracture.

The formulation of combinatorial differential forms, proposed by Forman for analysis of topologic... more The formulation of combinatorial differential forms, proposed by Forman for analysis of topological properties of discrete complexes, is extended by defining the operators required for analysis of physical processes dependent on scalar variables. The resulting description is intrinsic, different from the approach known as Discrete Exterior Calculus, because it does not assume the existence of smooth vector fields and forms extrinsic to the discrete complex. In addition, the proposed formulation provides a significant new modelling capability: physical processes may be set to operate differently on cells with different dimensions within a complex. An application of the new method to the heat/diffusion equation is presented to demonstrate how it captures the effect of changing properties of microstructural elements on the macroscopic behavior. The proposed method is applicable to a range of physical problems, including heat, mass and charge diffusion, and flow through porous media.

We present a novel and efficient parallel Newton-Krylov-Schur algorithm for the solution of the N... more We present a novel and efficient parallel Newton-Krylov-Schur algorithm for the solution of the Navier-Stokes equations. The governing equations are discretized using summation-by-parts operators of various orders, with boundary condition imposition and interface coupling achieved with the use of simultaneous approximation terms. For unsteady flows, the solution is integrated in time with explicit first stage, singly diagonally implicit RungeKutta methods of various orders. The discretized system of equations is solved through an inexact-Newton method with an approximate-Schur parallel preconditioner. The parallel capabilities of the algorithm can be leveraged to efficiently obtain steady solutions of complex turbulent flows, as well as to simulate unsteady transitional and turbulent flows based on implicit large-eddy and direct simulations.

HIGH-ORDER IMPLICIT NUMERICAL METHODS FOR UNSTEADY FLUID FLOW SIMULATION Pieter D. Boom <piete... more HIGH-ORDER IMPLICIT NUMERICAL METHODS FOR UNSTEADY FLUID FLOW SIMULATION Pieter D. Boom <pieter.boom@mail.utoronto.ca> Doctor of Philosophy Graduate Department of Aerospace Science and Engineering University of Toronto 2015 Unsteady computational fluid dynamics (CFD) is increasingly becoming a critical tool in the development of emerging technologies and modern aircraft. In spite of rapid mathematical and technological advancement, these simulations remain computationally intensive and time consuming. More efficient temporal integration will promote a wider use of unsteady analysis and extend its range of applicability. This thesis presents an investigation of efficient high-order implicit time-marching methods for application in unsteady compressible CFD. A generalisation of time-marching methods based on summation-by-parts (SBP) operators is described which reduces the number of stages required to obtain a prescribed order of accuracy, thus improving their efficiency. The cl...

Bulletin of the American Physical Society, 2017

these studies, we investigate the effect of mild synthetic jet actuation on a flat plate turbulen... more these studies, we investigate the effect of mild synthetic jet actuation on a flat plate turbulent boundary layer with the goal of interacting with the large scales in the log region of the boundary layer and manipulating the overall skin friction. Results will be presented from both large eddy simulations (LES) and wind tunnel experiments. In the experiments, a large parameter space of synthetic jet frequency and amplitude was studied with hot film sensors at select locations behind a pair of synthetic jets to identify the parameters that produce the greatest changes in the skin friction. The LES simulations were performed for a selected set of parameters and provide a more complete evaluation of the interaction between the boundary layer and synthetic jets. Five boundary layer thicknesses downstream, the skin friction between the actuators is generally found to increase, while regions of reduced skin friction persist downstream of the actuators. This pattern is reversed for forcing at low frequency. Overall, the spanwise-averaged skin friction is increased by the forcing, except when forcing at high frequency and low amplitude, for which a net skin friction reduction persists downstream. The physical interpretation of these results will be discussed.

This article presents constrained numerical optimization of fourth-order L-stable multistep Runge... more This article presents constrained numerical optimization of fourth-order L-stable multistep Runge-Kutta (MRK) methods. The methods are optimized relative to composite objective functions accounting for accuracy, internal stability, conditioning, and computational cost. Global stability properties and bounds on the coe cients are enforced through linear and nonlinear constraints. The relative bene ts of increasing the number of stages versus the number of steps is discussed, along with comparisons to implicit linear multistep (LM) and implicit Runge-Kutta (RK) methods. With the chosen objective function, the optimized MRK methods are not expected to be the most e cient. However, they do obtain a combination of properties that neither the LM or RK methods can. Furthermore, when applied to laminar ow over a circular cylinder, the optimized L-stable fourth-order two-step four-stage sti y-accurate singly-diagonally-implicit multistep Runge-Kutta method SDIMRK[4,2](4,2)L_SA_0 was the most...

Mathematics, 2021

The deformation of a solid due to changing boundary conditions is described by a deformation grad... more The deformation of a solid due to changing boundary conditions is described by a deformation gradient in Euclidean space. If the deformation process is reversible (conservative), the work done by the changing boundary conditions is stored as potential (elastic) energy, a function of the deformation gradient invariants. Based on this, in the present work we built a “discrete energy model” that uses maps between nodal positions of a discrete mesh linked with the invariants of the deformation gradient via standard barycentric coordinates. A special derivation is provided for domains tessellated by tetrahedrons, where the energy functionals are constrained by prescribed boundary conditions via Lagrange multipliers. The analysis of these domains is performed via energy minimisation, where the constraints are eliminated via pre-multiplication of the discrete equations by a discrete null-space matrix of the constraint gradients. Numerical examples are provided to verify the accuracy of the...

This article presents a comparison of the implicit (no model) large-eddy simulation (LES) techniq... more This article presents a comparison of the implicit (no model) large-eddy simulation (LES) technique and the local integral length-scale approximation (ILSA) sub lter model. The focus is on the numerical simulation of at plate turbulent boundary layers and active ow control using synthetic jets. After initial veri cation of the simulation setup, comparative studies are presented to investigate the dependence on spatial and temporal resolution, the level of arti cial dissipation, and some additional simulation parameters. Overall the ILSA model produces slightly more consistent and accurate results than the implicit LES approach. It also reduces computational cost by reducing the number of linear iterations required at each stage of the time-marching method. The controlled results generated by the implicit LES are in reasonably good agreement with experiment and give additional details of the spatial change in time-average skin friction. However, more work is needed to e ciently apply...

ArXiv, 2021

This paper presents an overview of high-order implicit time integration methods and their associa... more This paper presents an overview of high-order implicit time integration methods and their associated properties with a specific focus on their application to computational fluid dynamics. A framework is constructed for the development and optimization of general implicit time integration methods, specifically including linear multistep, Runge-Kutta, and multistep Runge-Kutta methods. The analysis and optimization capabilities of the framework are verified by rederiving methods with known coefficients. The framework is then applied to the derivation of novel singly-diagonally-implicit Runge-Kutta methods, explicit-first-stage singly-diagonally implicit Runge-Kutta methods, and singly-diagonallyimplicit multistep Runge-Kutta methods. The fourth-order methods developed have similar efficiency to contemporary methods; however a fifth-order explicit-first-stage singlydiagonally-implicit Runge-Kutta method is obtained with higher relative efficiency. This is confirmed with simulations of ...

Direct and Implicit Large-Eddy Simulation of the Taylor-Green Vortex Flow

To demonstrate the potential advantages of highorder spatial and temporal numerical methods, dire... more To demonstrate the potential advantages of highorder spatial and temporal numerical methods, direct numerical and implicit large-eddy simulation of the Taylor-Green vortex flow is computed using a variable-order finite-difference code on multi-block structured meshes. The spatial operators satisfy the summation-by-parts property, with block interfaces and boundary conditions enforced with simultaneousapproximation-terms. The solution is integrated in time with explicit-first-stage, singly-diagonallyimplicit Runge-Kutta methods. An investigation into artificial dissipation and spatial filtering shows filtering is much more computationally efficient at moderate Courrant numbers, however, it does eventually place a limit on the time step. Grid convergence studies show excellent performance of higher resolution simulations, accurately capturing the decay of kinetic energy with a decay proportional to t−2 after transition to turbulence. The simulations also produce very good energy spect...

International Journal of Solids and Structures, 2021

A direct formulation of linear elasticity of cell complexes based on discrete exterior calculus i... more A direct formulation of linear elasticity of cell complexes based on discrete exterior calculus is presented. The primary unknown are displacements, represented by primal vector-valued 0-cochain. Displacement differences and internal forces are represented by primal vectorvalued 1-cochain and dual vector-valued 2-cochain, respectively. The macroscopic constitutive relation is enforced at primal 0-cells with the help of musical isomorphisms mapping cochains to smooth fields and vice versa. The balance of linear momentum is established at primal 0-cells. The governing equations are solved as a Laplace equation with a non-local and non-diagonal material Hodge star. Numerical simulations of several classical problems with analytic solutions are presented to validate the formulation. Good agreement with known solutions is obtained. The formulation provides a method to calculate the relations between displacement differences and internal forces for any lattice structure, when the structure is required to follow a prescribed macroscopic elastic behaviour. This is also the first and critical step in developing formulations for dissipative processes in cell complexes.

Applied Sciences, 2021

We investigated the derivation of numerical methods for solving partial differential equations, f... more We investigated the derivation of numerical methods for solving partial differential equations, focusing on those that preserve physical properties of Hamiltonian systems. The formulation of these properties via symplectic forms gives rise to multisymplectic variational schemes. By using analogy with the smooth case, we defined a discrete Lagrangian density through the use of exponential functions, and derived its Hamiltonian by Legendre transform. This led to a discrete Hamiltonian system, the symplectic forms of which obey the conservation laws. The integration schemes derived in this work were tested on hyperbolic-type PDEs, such as the linear wave equations and the non-linear seismic wave equations, and were assessed for their accuracy and the effectiveness by comparing them with those of standard multisymplectic ones. Our error analysis and the convergence plots show significant improvements over the standard schemes.

Journal of Computational Physics, 2018

This article presents constrained numerical optimization of high-order linearly and algebraically... more This article presents constrained numerical optimization of high-order linearly and algebraically stable diagonally-implicit Runge-Kutta methods. After satisfying the desired order conditions, undetermined coe cients are optimized with respect to objective functions which consider accuracy, stability, and computational cost. Constraints are applied during the optimization to enforce stability properties, to ensure a well-conditioned method, and to limit the domain of the abscissa. Two promising third-order methods are derived using this approach, labelled SDIRK[3,(1,2,2)](3)L 14 and SDIRK[3,1](4)L SA 5. Both optimized schemes have a good balance of properties. The relative error norm of the latter, the L 2 -norm scaled by a function of the number of implicit stages, is a factor of two smaller than comparable methods found in the literature. Variations on these methods are discussed relative to trade-o↵s in their accuracy and stability properties. A novel fifthorder scheme SDIRK[5,1](5)L 02 is derived with a significantly lower relative error norm than the comparable fifth-order A-stable reference method. In addition, the optimized scheme is L-stable. The accuracy and relative e ciency of the Runge-Kutta methods are verified through numerical simulation of van der Pol's equation, as well as numerical simulation of vortex shedding in the laminar wake of a circular cylinder, and in the turbulent wake of a NACA 0012 airfoil. These results demonstrate the value of numerical optimization for selecting undetermined coe cients in the construction of high-order ⇤

Journal of Computational Physics, 2017

Combined with simultaneous approximation terms, summation-by-parts (SBP) operators o↵er a versati... more Combined with simultaneous approximation terms, summation-by-parts (SBP) operators o↵er a versatile and e cient methodology that leads to consistent, conservative, and provably stable discretizations. However, diagonal-norm operators with a repeating interior-point operator that have thus far been constructed su↵er from a loss of accuracy. While on the interior, these operators are of degree 2p, at a number of nodes near the boundaries, they are of degree p, and therefore of global degree p -meaning the highest degree monomial for which the operators are exact at all nodes. This implies that for hyperbolic problems and operators of degree greater than unity they lead to solutions with a global order of accuracy lower than the degree of the interior-point operator. In this paper, we develop a procedure to construct diagonal-norm first-derivative SBP operators that are of degree 2p at all nodes and therefore can lead to solutions of hyperbolic problems of order 2p+1. This is accomplished by adding nonzero entries in the upper-right and lower-left corners of SBP operator matrices with a repeating interior-point operator. This modification necessitates treating these new operators as elements, where mesh refinement is accomplished by increasing the number of elements in the mesh rather than increasing the number of nodes. The significant improvements in accuracy of this new family, for the same repeating interior-point operator, are demonstrated in the context of the linear

SIAM Journal on Scientific Computing, 2015

This article extends the theory of classical finite-difference summation-by-parts (FD-SBP) timema... more This article extends the theory of classical finite-difference summation-by-parts (FD-SBP) timemarching methods to the generalized summation-by-parts (GSBP) framework. Dual-consistent GSBP time-marching methods are shown to retain: A and L-stability, as well as superconvergence of integral functionals when integrated with the quadrature associated with the discretization. This also implies that the solution approximated at the end of each time step is superconvergent. In addition GSBP time-marching methods constructed with a diagonal norm are BN-stable. This article also formalizes the connection between FD-SBP/GSBP time-marching methods and implicit Runge-Kutta methods. Through this connection, the minimum accuracy of the solution approximated at the end of a time step is extended for nonlinear problems. It is also exploited to derive conditions under which nonlinearly stable GSBP time-marching methods can be constructed. The GSBP approach to time marching can simplify the construction of high-order fully-implicit Runge-Kutta methods with a particular set of properties favourable for stiff initial value problems, such as L-stability. It can facilitate the analysis of fully discrete approximations to PDEs and is amenable to to multi-dimensional spcae-time discretizations, in which case the explicit connection to Runge-Kutta methods is often lost. A few examples of known and novel Runge-Kutta methods associated with GSBP operators are presented. The novel methods, all of which are L-stable and BN-stable, include a four-stage seventh-order fully-implicit method, a three-stage third-order diagonally-implicit method, and a fourth-order four-stage diagonally-implicit method. The relative efficiency of the schemes is investigated and compared with a few popular non-GSBP Runge-Kutta methods.

55th AIAA Aerospace Sciences Meeting, 2017

This paper presents a numerical investigation of the tradeo↵s between various discretization appr... more This paper presents a numerical investigation of the tradeo↵s between various discretization approaches and operators, based on diagonal-norm summation-by-parts (SBP) operators, using the two-dimensional linear convection equation and simultaneous approximation terms (SATs) for the weak imposition of boundary conditions and interface coupling. In particular, it focuses on operators which include boundary nodes. Of the operators considered, the hybrid-Gauss-trapezoidal-Lobatto SBP operators are the most e cient. Little di↵erence in e ciency is observed between the divergence and skew-symmetric forms, making the latter preferred given its provable stability on curved meshes. The traditional finite-di↵erence refinement strategy is the most e cient, and the discontinuous element approach the least. The continuous element refinement strategy has comparable e ciency to the traditional approach when not exhibiting lower convergence rates. This motivates a hybrid approach whereby discontinuous elements are constructed from continuous subelements. This hybrid approach is found to inherit the higher convergence rates of the traditional and discontinuous approaches, and higher e ciency relative to the discontinuous approach.