I work on the development, analysis and implementation of fast, reliable, and accurate numerical methods for the solution of local and nonlocal models from science and engineering at different scales (e.g., macro and nanoscale). In my research, I use and develop meshfree methods, finite element, and isogeometric schemes to simulate the mechanical response of materials and solid structures, including in extreme conditions (e.g., fragmentation, fracture, and large deformation).
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Efficient meshfree modeling of extreme deformations via Lagrangian/semi-Lagrangian coupling
Efficient meshfree modeling of extreme deformations via Lagrangian/semi-Lagrangian coupling
In many civil and military applications (e.g., for a safe design), it is necessary to predict structural response efficiently and accurately under extreme loading conditions, large deformations, and material failure, using computational tools. Meshfree methods have been shown to be effective for this class of problems. In particular, the semi-Lagrangian (SL) reproducing kernel particle method (RKPM) has been proved to be suited for problems involving damage and fragmentations. This approach can handle the radical changes in topology present in these problems by reconstructing the field approximations based on the current configuration (i.e., at every time step of the simulation). However, this results in a very high computational cost. The Lagrangian (L) version of RKPM, instead, constructs the approximations only once in the reference configuration and is therefore computationally advantageous, but it breaks down when the topology of the problem changes (e.g., during fragmentation). To retain the advantages of both formulations, we developed a coupling scheme to transition between the two by spatially blending the Lagrangian and semi-Lagrangian RK shape functions and shape function gradients. This way we can employ the more costly semi-Lagrangian approach only where and when necessary. We showed that the resulting coupled scheme retains the same temporal stability as the classical schemes, and that there isn’t any wave reflection at the transition zones. We verified through a suite of numerical problems involving large deformations, plasticity, impacts, perforations, and fragmentations, that the proposed coupling scheme can achieve the same accuracy of a full semi-Lagrangian RK simulation at a fraction of the computational cost.
Meshfree RK discretization and kernel function.
Taylor bar simulation. Top: L; middle: SL; bottom: coupled.
Impact-perforation simulation setup of bullet impacting a steel plate.
High-velocity impact of a bullet perforating a steel plate. Left: semi-Lagrangian RK. Right: coupled approach.
Related publications:
A Lagrangian/semi-Lagrangian coupling approach for accelerated meshfree modelling of extreme deformation problems, Computer Methods in Applied Mechanics and Engineering, 381, (2021).
Collaborators: J.S. Chen (UCSD), Jesse Sherburn (U.S. Army ERDC), Michael J. Roth (U.S. Army ERDC), Jonghyuk Baek (UCSD), Haoyan Wei (Ansys).
Continuum modeling for nanoscale strain engineering with nonlocal interactions
Continuum modeling for nanoscale strain engineering with nonlocal interactions
Semiconductors are employed in various kinds of electronic devices. The goal of nanoscale strain engineering is to tune the electronic properties of semiconductors to enhance their performance, by modulating their nanoscale strain field. This can be achieved via the use of the Van der Waals forces resulting from the nonlocal interaction between different materials. To simulate these interactions and the resulting strain fields, molecular dynamic (MD) simulations can be employed. However, the discrete nature of MD limits the simulation scale and results in a high computational cost, making it unsuitable to test the effect of using various materials and/or different geometrical configurations. Therefore, our goal is to develop an efficient continuum approach to simulate these systems. In our work, we are considering layers of MoS2 with a multi-hole Si3N4 substrate.
AFM data representing the shape of the periodic multi-hole Si3N4 substrate. (courtesy of E. Tadmor and M.-K. Choi).
Experimental bend contours of the MoS2 layer obtained via dark-field imaging. (courtesy of E. Tadmor and M.-K. Choi).
Continuum simulation result for the deformed MoS2 layer interacting with a periodic multi-hole Si3N4 substrate.
Collaborators: David Kamensky (UCSD), Ellad Tadmor (University of Minnesota), Zhaoxiang Shen (UCSD), Moon-Ki Choi (University of Minnesota).
High-order RKPM meshfree nonlocal peridynamic modeling
High-order RKPM meshfree nonlocal peridynamic modeling
Peridynamics (PD) is a nonlocal reformulation of the local mechanics theory, based on an integral operator on a spherical ball, the radius of which – the PD horizon – defines the length-scale of the nonlocality. Therefore, peridynamic models do not have differentiability requirements on the displacement fields. Since its introduction in 2000, peridynamics has been receiving a lot of attention due to its capability of simulating autonomous crack growth without having to explicitly track the crack interface but has since been employed in a variety of different problems (e.g., nonlocal diffusion, upscaling of molecular dynamics). Due to its simplicity, the most common discretization method for peridynamic models used in engineering problems is a node-based meshfree approach. This method discretizes peridynamic domains by a set of nodes, each associated with a nodal cell with a characteristic volume, leading to a particle-based description of continuum systems. The behavior of each particle is then considered representative of its cell. This limits the convergence rate to the first order. The goal of this work was to increase the convergence rate of meshfree peridynamic simulations. This was achieved via the use of reproducing kernel (RK) shape functions for the approximation of the displacement field, which allows arbitrary completeness and smoothness of the approximation. Furthermore, inaccurate numerical integration leads to oscillations in the convergence behavior of the numerical solution. This was solved by the uses of Gauss quadrature, mapped to the spherical balls. The ability of this RK enhanced peridynamic approach to achieve high-order convergence was demonstrated through the solution of several one-dimensional and two-dimensional problems.
Quadrature over the nonlocal interaction balls. Left: two-dimensional Gauss quadrature mapped to polar coordinates; right: quadrature points mapped to retain angular uniformity.
Convergence for a manufactured cubic solution using linear and quadratic RK approximation: rates higher than first-order are achieved.
Related publications:
A reproducing kernel enhanced approach for peridynamic solutions. Computer Methods in Applied Mechanics and Engineering, 340, 1044-1078.
Collaborators: Yu Leng (Purdue), J.S. Chen (UCSD), John Foster (UT Austin), Pablo Seleson (ORNL).
A waveform relaxation Newmark method for fast elastodynamic simulations
A waveform relaxation Newmark method for fast elastodynamic simulations
For structural dynamic problems, governed by a second-order hyperbolic system of ordinary differential equations, the Newmark family of methods is one of the most well-known and widely used families of direct integration methods. Its implicit implementation is unconditionally stable but requires the solution of a linear system, which makes it computationally expensive; its explicit form, on the other hand, has a low computational cost as it doesn’t require the solution of systems but is conditionally stable, thus limiting the allowed time increment. Our goal was to develop a time-integration scheme that retains the unconditional stability of implicit Newmark at a computational cost comparable to that of explicit schemes. To this end, we paired the waveform relaxation iterative schemes, originally developed for first-order systems, with the implicit form of the Newark method to form a Waveform Relaxation (WR) Newmark algorithm. The main steps of the algorithm are system partitioning obtained by splitting of the mass, damping, and stiffness matrices to extract their diagonal (Jacobi scheme) or lower diagonal (Gauss-Seidel scheme) parts; the development of an iterative waveform relaxation scheme; the solution with the Newmark implicit scheme. Due to partitioning, the resulting iterative scheme does not require the inversion of a full matrix. Therefore, although it requires iterations in the time domain, since convergence can be obtained quickly by using time window sizes, it requires significantly lower computational costs compared to implicit Newark, which is emphasized for larger problem sizes. This was shown via several numerical examples. Furthermore, the Waveform Relaxation Newmark algorithm retains unconditional stability; it is unstructured in time domain and well suited for time parallelization compared with the inherently sequential Newmark method; and it is well suited for multi-rate integration of subcomponents (such as in bimaterial problems).
Data flow: Jacobi scheme.
Data flow: Gauss-Seidel scheme.
CPU time for a two-dimensional plate discretized with finite elements. As the system size increases, the solution time is lower for the WR Newmark approach compared to classical implicit Newmark.
Related publications:
A waveform relaxation Newmark method for structural dynamics problems. Computational Mechanics, 63, 1223-1242.
Collaborators: Haim Waisman (Columbia University), J.S. Chen (UCSD)
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