Current focus
My group's current focus is on
Other current interests include image-based modeling of microstructures from CT-scans, with application to most of the aforementioned topics.
- fundamental development of numerical methods, in particular meshfree methods (accurate, convergent, stable, efficient, high-order, flexible and adaptive discretizations)
- unifying frameworks for meshfree and particle methods (local and non-local, continuum and discrete)
- computational fracture mechanics
- thermomechanics with application to viscous problems and phase change (thermo-elastic, visco-elastic, visco-thermo-elastic), and
- novel formulations for composites.
Other current interests include image-based modeling of microstructures from CT-scans, with application to most of the aforementioned topics.
High-order Accurate Galerkin Meshfree Methods
Variational Consistency
While approximations with arbitrary-order accuracy can easily be constructed, this does not guarantee that the order of accuracy will be achieved in the Galerkin solution to a problem, even in the simple setting of a linear displacement field, or a so-called patch test. While not a difficultly in traditional approaches, in meshfree methods the numerical integration of rational approximations with no intrinsic tie of their support to integration cells makes this difficult to achieve. As a result, high-order quadrature in traditional Gaussian quadrature is necessary to ensure stable and accurate solutions, yet the associated computational cost makes this intractable. Low-order integration is much faster of course, but cannot yield the accuracy of the former, presenting an impasse. In (Chen, Hillman and Rüter, 2013) the concept of variationally consistent integration (VCI) was introduced, where the exact requirements for nth-order accuracy in the Galerkin equation were set forth. This work finally showed an alternative path to high-order accuracy without high-order quadrature. An efficient correction was then introduced where optimal convergence can be achieved with low-order integration, alleviating the efficiency bottleneck that has plagued Galerkin-based meshfree methods since their inception in the 1990s. This work has since served as the basis of new methods developed by other research groups. A two-part paper (preprints: Hillman and Lin 2021a, Hillman and Lin 2021b) extends VCI to thermomechanical problems, where the nth order conditions are derived for a two-field problem for the first time.
While this approach can provide significant speedups, the remaining cost relative to traditional FEM and low-order meshfree methods like smoothed particle hydrodynamics (SPH) still makes these methods less attractive. In the future, Hillman plans to accelerate the computations even further by leveraging recent peridynamic developments: in (Jafarzadeh et al., 2021a), a fast-Fourier transform method was introduced that circumvents the traditional limitations of periodic domains (Jafarzadeh et al., 2021a) and linear problems (preprint: Jafarzadeh et al., 2021b). This technique allows speedups of orders of magnitude (hours to minutes, days to hours).
While approximations with arbitrary-order accuracy can easily be constructed, this does not guarantee that the order of accuracy will be achieved in the Galerkin solution to a problem, even in the simple setting of a linear displacement field, or a so-called patch test. While not a difficultly in traditional approaches, in meshfree methods the numerical integration of rational approximations with no intrinsic tie of their support to integration cells makes this difficult to achieve. As a result, high-order quadrature in traditional Gaussian quadrature is necessary to ensure stable and accurate solutions, yet the associated computational cost makes this intractable. Low-order integration is much faster of course, but cannot yield the accuracy of the former, presenting an impasse. In (Chen, Hillman and Rüter, 2013) the concept of variationally consistent integration (VCI) was introduced, where the exact requirements for nth-order accuracy in the Galerkin equation were set forth. This work finally showed an alternative path to high-order accuracy without high-order quadrature. An efficient correction was then introduced where optimal convergence can be achieved with low-order integration, alleviating the efficiency bottleneck that has plagued Galerkin-based meshfree methods since their inception in the 1990s. This work has since served as the basis of new methods developed by other research groups. A two-part paper (preprints: Hillman and Lin 2021a, Hillman and Lin 2021b) extends VCI to thermomechanical problems, where the nth order conditions are derived for a two-field problem for the first time.
While this approach can provide significant speedups, the remaining cost relative to traditional FEM and low-order meshfree methods like smoothed particle hydrodynamics (SPH) still makes these methods less attractive. In the future, Hillman plans to accelerate the computations even further by leveraging recent peridynamic developments: in (Jafarzadeh et al., 2021a), a fast-Fourier transform method was introduced that circumvents the traditional limitations of periodic domains (Jafarzadeh et al., 2021a) and linear problems (preprint: Jafarzadeh et al., 2021b). This technique allows speedups of orders of magnitude (hours to minutes, days to hours).
Figure: VC- correction of Gaussian integration (GI), direct nodal integration (DNI), stabilized non-conforming nodal integration (SNNI) and comparisons to stabilized conforming nodal integration (SCNI).
Figure: VC- correction of naturally stabilized nodal integration (NSNI) in thermo-mechanical problems for coupled and uncoupled cases (delta = 1 and 0 respectively).
Consistent Weak Formulations
Meshfree methods are unique in both their abilities, but also their properties concerning the Galerkin solution. The above work addresses the issue of solution error due to quadrature; however, these methods also are non-interpelotory, so that essential (Dirichlet) boundary conditions are not easily imposed, which has been the subject of research since the mid-1990s. This problem has been addressed by either weak enforcement, or more commonly, strong enforcement in conjunction with the traditional weak formulation. Yet, in the latter more popular approach, there remained an inconsistency yet to be investigated: the subspaces required in the standard weak formulation cannot be constructed using most meshfree approximations.
In recent work (Hillman and Lin, 2021), my Ph.D. student Lin and I showed that this inconsistency resulted in an O(h) (with h the nodal spacing) error in the energy norm of the Galerkin solution to second-order PDEs, independent of the actual order of approximation employed, placing a barrier on the order of Galerkin accuracy to linear. This inconsistency was also shown to result in the loss of the energy norm projection of the error onto the test space, no longer yielding the celebrated best approximation property in Galerkin methods. To address this inconsistency, Hillman and Lin introduced consistent weak forms, which permitted the use of meshfree approximations. It was shown that this formulation restores the projection property of the Galerkin solution, and allows p-refinement, pth order optimal rates in h-refinement, and introduced a new concept called a-refinement. Future work in this area includes the introduction of the variational multiscale method (VMS) for alternative effective boundary condition enforcement, with VMS then developed for other constrained problems in meshfree methods.
Meshfree methods are unique in both their abilities, but also their properties concerning the Galerkin solution. The above work addresses the issue of solution error due to quadrature; however, these methods also are non-interpelotory, so that essential (Dirichlet) boundary conditions are not easily imposed, which has been the subject of research since the mid-1990s. This problem has been addressed by either weak enforcement, or more commonly, strong enforcement in conjunction with the traditional weak formulation. Yet, in the latter more popular approach, there remained an inconsistency yet to be investigated: the subspaces required in the standard weak formulation cannot be constructed using most meshfree approximations.
In recent work (Hillman and Lin, 2021), my Ph.D. student Lin and I showed that this inconsistency resulted in an O(h) (with h the nodal spacing) error in the energy norm of the Galerkin solution to second-order PDEs, independent of the actual order of approximation employed, placing a barrier on the order of Galerkin accuracy to linear. This inconsistency was also shown to result in the loss of the energy norm projection of the error onto the test space, no longer yielding the celebrated best approximation property in Galerkin methods. To address this inconsistency, Hillman and Lin introduced consistent weak forms, which permitted the use of meshfree approximations. It was shown that this formulation restores the projection property of the Galerkin solution, and allows p-refinement, pth order optimal rates in h-refinement, and introduced a new concept called a-refinement. Future work in this area includes the introduction of the variational multiscale method (VMS) for alternative effective boundary condition enforcement, with VMS then developed for other constrained problems in meshfree methods.
Figure: Consistent strong enforcement of boundary conditions: consistent weak forms (CWFs) restore the projection of the Galerkin error onto the test space and ability to provide optimal rates of convergence.
Figure: CWFs enable a new concept called "a-refinement": increasing the kernel measure can lower the error by an order of magnitude.
High-order Peridynamics and Shock-enriched Meshfree Methods
Also in recent work, Hillman, Pasetto, and Zhou, 2019, investigated the relationship between the traditional meshfree methods and peridynamic-based meshfree methods that have the potential to tackle the long-standing challenge of computational fracture mechanics. While some relationships had been established in ideal conditions, this work finally illuminated the precise connection between the two methods. Casting the approximations as a unified operator, both classical and peridynamic meshfree derivatives can be obtained. This work also generalized the concept of a peridynamic pseudo-derivative and allowed both high-order numerical differentiation under peridynamics and high-order accurate solutions. This was then leveraged to obtain high-order accurate (convergent) peridynamics, whereas the previously existing formulation was also shown to provide no convergence of the solution. Exposing the link between classical and peridynamic meshfree methods led to Hillman's NSF CAREER award in 2020. Other recent work in this area includes a particle-based shock method that precludes Gibb's instability in shock problems and embeds shock physics (Zhou and Hillman, 2020), while a similar particle-based scheme was also developed for traditional meshfree methods in Baek et al., The introduction of other advances in the classical meshfree methods into peridynamics is also planned for robust simulation of fracture initiation and propagation.
Also in recent work, Hillman, Pasetto, and Zhou, 2019, investigated the relationship between the traditional meshfree methods and peridynamic-based meshfree methods that have the potential to tackle the long-standing challenge of computational fracture mechanics. While some relationships had been established in ideal conditions, this work finally illuminated the precise connection between the two methods. Casting the approximations as a unified operator, both classical and peridynamic meshfree derivatives can be obtained. This work also generalized the concept of a peridynamic pseudo-derivative and allowed both high-order numerical differentiation under peridynamics and high-order accurate solutions. This was then leveraged to obtain high-order accurate (convergent) peridynamics, whereas the previously existing formulation was also shown to provide no convergence of the solution. Exposing the link between classical and peridynamic meshfree methods led to Hillman's NSF CAREER award in 2020. Other recent work in this area includes a particle-based shock method that precludes Gibb's instability in shock problems and embeds shock physics (Zhou and Hillman, 2020), while a similar particle-based scheme was also developed for traditional meshfree methods in Baek et al., The introduction of other advances in the classical meshfree methods into peridynamics is also planned for robust simulation of fracture initiation and propagation.
Application of methods to natural and man-made disasters
Disasters, whether naturally occurring such as earthquakes, landslides, tsunamis, and storms, or man-made, e.g. those resulting from terrorism, contribute to a significant loss of life and property nationally and globally. Death tolls from one single natural disaster can range in the hundreds of thousands; cumulative tolls for man-made disasters (such as those during the war on terror) can be on the same magnitude. Effective numerical methods facilitate the advancement of protection against these disasters. My primary research focus is the development of robust and efficient numerical methods for extremely large deformation problems such as blast, fragment-impact processes and systems and components undergoing extreme conditions.
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Perforation of a concrete block using a stabilized meshfree method
Numerical and experimental crater and hole
comparison of fragmentation patterns in a perforation event
comparison of lateral cracking
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I have developed a computational formulation called variationally consistent integration [1,2] to achieve high accuracy with low order domain integration for meshfree methods such as the reproducing kernel particle method (RKPM) [3,4], which alleviates the efficiency bottleneck associated with quadrature. It enables accurate and efficient simulations of material damage processes and structural failure, earth moving, damage assessment of landslides, and structures subjected to man-made and natural disasters. This research has been implemented into large-scale computational codes used by the Army ERDC, and Sandia National Laboratories, to model bullet penetration into geomaterials (concrete, soil) to numerically investigate damage mechanisms in these events, and facilitate advancement in protective structures. Simulation results have been validated under blind test prediction for various sizes of targets, projectile sizes, and velocities (figures to the left). Slope stability analyses (figure below), as well as simulations of full scale earth moving for Caterpillar have also been performed using the stabilized RKPM method J.S. Chen and I have developed [2,5-7].
Triggering of landslide due to 1989 Loma Prieta Earthquake
Subsequently, I developed an accelerated meshfree method [6], that exhibits high accuracy, stability, and very low computational cost (20x faster than previous methods), with excellent performance in large deformation problems involving fragment-impact and penetration, blast-loaded structures, slope stability, and metal forming problems (figures below).
comparison of cone cracking
comparison of radial cracking
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Validation of landslide simulation
Landslide simulation with progressive failure
Perforation problem with and without stabilization (tensile damage)
Steel plate subject to blast loading from the UCSD extreme event simulator
Overview and technical summary (2 articles)
Technical summaries of the theory and application of these works can be found in the articles below:
Technical summaries of the theory and application of these works can be found in the articles below:
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References
[1] Chen, J. S., Hillman, M., Ruter, M., An arbitrary order variationally consistent method for Galerkin meshfree methods, International Journal for Numerical Methods in Engineering, Vol. 95, pp. 387-418, 2013, doi:10.1002/nme.4512.
[2] Hillman, M., Chen, J. S., Stabilized and variationally consistent nodal integration for meshfree modeling of impact problems, Computational Particle Mechanics, Vol. 1, pp. 245-256, 2014, doi:10.1007/s40571-014-0024-5.
[3] Liu, W. K., Jun, S., Zhang, Y. F.. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, Vol. 20, pp. 1081-1106, 1995, doi: 10.1002/fld.1650200824.
[4] Chen, J.S., Pan, C., Wu, C.T., Liu, W. K. Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures. Computer Methods in Applied Mechanics and Engineering, Vol 139, pp. 195–227, 1996, doi:10.1016/S0045-7825(96)01083-3.
[5] Sherburn, J.A.,Roth M.J. , Chen, J.S., Hillman, M., Meshfree modeling of concrete slab perforation using a reproducing kernel particle impact and penetration formulation, Int. J. Impact Eng. 86 (2015) 96–110. doi:10.1016/j.ijimpeng.2015.07.009.
[6] Hillman, M., Chen, J. S., An accelerated, convergent and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics, 2015, International Journal for Numerical Methods in Engineering, doi:10.1002/nme.5183.
[1] Chen, J. S., Hillman, M., Ruter, M., An arbitrary order variationally consistent method for Galerkin meshfree methods, International Journal for Numerical Methods in Engineering, Vol. 95, pp. 387-418, 2013, doi:10.1002/nme.4512.
[2] Hillman, M., Chen, J. S., Stabilized and variationally consistent nodal integration for meshfree modeling of impact problems, Computational Particle Mechanics, Vol. 1, pp. 245-256, 2014, doi:10.1007/s40571-014-0024-5.
[3] Liu, W. K., Jun, S., Zhang, Y. F.. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, Vol. 20, pp. 1081-1106, 1995, doi: 10.1002/fld.1650200824.
[4] Chen, J.S., Pan, C., Wu, C.T., Liu, W. K. Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures. Computer Methods in Applied Mechanics and Engineering, Vol 139, pp. 195–227, 1996, doi:10.1016/S0045-7825(96)01083-3.
[5] Sherburn, J.A.,Roth M.J. , Chen, J.S., Hillman, M., Meshfree modeling of concrete slab perforation using a reproducing kernel particle impact and penetration formulation, Int. J. Impact Eng. 86 (2015) 96–110. doi:10.1016/j.ijimpeng.2015.07.009.
[6] Hillman, M., Chen, J. S., An accelerated, convergent and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics, 2015, International Journal for Numerical Methods in Engineering, doi:10.1002/nme.5183.
Application of methods to earth-moving problems
These methods can be applied to a variety of problems involving large/extreme deformations. An example is shown in the following video.
Some LS-DYNA simulations using NSNI and Quasi-linear RKPM
LS-DYNA (*Control_SPH keyword, set FORM = 12): NSNI and quasi-linear RKPM simulation of soil moving.
Some details from LSTC (now ANYSIS):
Moving Least-Squares based formulation is used to model large deformations of cohesive soil. SPH simulation performed in LS-Dyna, surface generated in Paraview, and rendered in Blender through VisualSPHysics. In order to find the input decks, please visit : http://www.dynaexamples.com/ See all DYNA conference publications at : http://www.dynalook.com/
Paper is here:
Yreux, Edouard. "MLS-based SPH in LS-DYNA® for Increased Accuracy and Tensile Stability." 15th International LS-DYNA® Users Conference. 2018.
Some details from LSTC (now ANYSIS):
Moving Least-Squares based formulation is used to model large deformations of cohesive soil. SPH simulation performed in LS-Dyna, surface generated in Paraview, and rendered in Blender through VisualSPHysics. In order to find the input decks, please visit : http://www.dynaexamples.com/ See all DYNA conference publications at : http://www.dynalook.com/
Paper is here:
Yreux, Edouard. "MLS-based SPH in LS-DYNA® for Increased Accuracy and Tensile Stability." 15th International LS-DYNA® Users Conference. 2018.
More comparisons of enhanced RKPM via LS-DYNA: