A Lagrangian gradient smoothing method for solid-flow problems using simplicial mesh


Recent technological advances have seen the evolution of a broad range of numerical methods for solving various types of engineering problems. Most notably, the mesh free method: which is not prone to meshing issues hence preferable in handling extremely large deformation problems. Specifically, smoothed particle hydrodynamics stands out among the mesh-free methods as a robust Lagrangian particle method due to its versatility in application. Unfortunately, this technique possesses some inherent drawbacks in terms of stability and accuracy aspects, which for a long time have been a bottleneck for its wider applications. Regardless, many corrections to these instability issues have been developed, however, they more often introduce additional problems. Therefore, there is a general desire to develop an alternative technique that will help solve “solid- flow” problems without any inherent shortcomings such as the tensile instability that has been existing in the smoothed particle hydrodynamics technique since it was first conceived.

Professor G.R. Liu and his PhD student Zirui Mao from the Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati developed a novel Lagrangian gradient smoothing method (L-GSM) that utilized the gradient smoothing technique to approximate the gradient of the field variables, based on the standard gradient smoothing method which works well with Euler grids for general fluids. This was inspired by the excellent performance of the recently introduced Eulerian gradient smoothing method for fluid flows. Their work is currently published in the research journal, International Journal for Numerical Methods in Engineering.

The research method employed commenced with the adoption of the Delaunay triangulation algorithm for use in updating the connectivity of the particles, in that the supporting neighboring particles could be determined for accurate gradient approximations. Next, the research pair devised special techniques for the treatment of the three types of boundaries encountered. They then proceeded to develop an advanced gradient smoothing method operation so as to ensure better consistency. Eventually, the developed L-GSM was evaluated for tensile stability conditions as well as validated using benchmarking examples of incompressible flows and applied to solve a practical problem of solid flows.

After a comparison between the numerical results with theoretical solutions, it was observed that the L-GSM scheme yielded very precise result for all the cross-examined examples. Additionally, the researchers noted that the computation efficiency of L‐GSM was significantly higher than that of the smoothed particle hydrodynamics method, especially when a large number of particles were used. Regardless, this technique was also seen to experience an undesirable computational efficiency with the current low‐efficient nearest neighboring particle searching algorithm of Delaunay triangulation.

The Liu-Mao study has presented the development of a novel L-GSM for solid-flow problems in Lagrangian system based on Eulerian gradient smoothing method. In this work, improvements have been made in the gradient smoothing operation on the boundary of the problem domain to ensure the consistence not only inside the problem domain but also on the boundary. It has been observed that both the theoretical analysis and the numerical tests demonstrate that the L-GSM is of first-order accuracy for all particles including those on boundary. All in all, this novel L-GSM approach is superior to the smoothed particle hydrodynamics method in stability and efficiency, even in accuracy sometimes for solid-like flows. More so, this technique offers a valuable alternative to the Eulerian gradient smoothing method for fluid-flow problems.


A Lagrangian gradient smoothing method for solid-flow problems using simplicial mesh. Advances in Engineering


About the author

Zirui Mao received his B.S. in Physics from Hebei Normal University in 2010 and M.E. in Solid Mechanics from Tianjin University in 2013. He is current a Ph.D. candidate majored in computational mechanics in University of Cincinnati.

His research focus on improving the existing meshfree methods such as SPH and developing novel meshfree methods to handle the extremely large deformation problems that are difficult to dealt with by the traditional grid-based methods. The current research is simulating the earthquake-induced fluid-like landslides under the gravity and ground motion due to the earthquake using the proposed Lagrangian Gradient Smoothing Method. Meanwhile, He is also working part-time at UC simulation center and P&G to develop a series of theoretical and numerical models predicting the mechanical properties and behaviors of the polymer materials under the squeeze flow and hot ultrasonic bonding process and predicting the bond strength of the designed patterns by P&G designers. Previous works include the Applications in Soil column collapse/Landslides/free surface flows/fluid flows/electrostatic transport of lunar dust/heat transfer/squeeze flow/hot ultrasonic bonding/strain to failure using the numerical methods of L-GSM/SPH/PIC and by the theoretical models.

About the author

Dr. GR Liu is currently the Professor and Ohio Eminent Scholar working in the department of Aerospace Engineering and Engineering Mechanics in University of Cincinnati. He is also the editor-in-chief of ‘International journal of computational methods’, associate editors of ‘Inverse Problems in Science&Engineering’, ‘Microfluidics and Nanofluidics’ et.al.

Dr. Liu is one of the key founders of smoothed finite element method and the founder of a series of novel numerical methods, such as gradient smoothing method. He has hundreds of journal and conference paper publications and more than 10 published famous and popular books.

Dr. Liu is the head of the GRLab named ‘Computation for Sustainability’ with the mission to develop advanced computational technology for solving complex problems in engineering and science, and to provide computational solutions to the current and future industry. The applications include but not limited in Fluid-Structure interaction, Rotorcraft, fragmentation, processing, Biomechanics, blood flow, generalized fluid flow, penetration, high velocity impact, explosion, Membranes, twisting and punching of solids, fluid-like landslides by a series of in-house codes.


Zirui Mao, G. R. Liu. A Lagrangian gradient smoothing method for solid‐flow problems using simplicial mesh. International Journal for Numerical Methods in Engineering, volume 113 (2018);pages 858–890

Go To International Journal for Numerical Methods in Engineering

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