me563 project final - psu compbio

D e p a r t m e n t o f M e c h a n i c a l a n d N u c l e a r E n g i n e e r i n g , U n i v e r s i t y P a r k , P A

TheDevelopmentandAnalysisofPorcineTrabecularBoneGroupMembers:ZiwenFang

Spring2016

08 Fall

ME563:Nonlinear

FiniteElementAnalysis

ASemesterReporton:

TheDevelopmentandAnalysisofPorcineTrabecularBone

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TableofContentsTableofContents...................................................................................................................2ExecutiveSummary................................................................................................................3Acknowledgements................................................................................................................4ListofFigures.........................................................................................................................5Section1:BackgroundandProjectPlan..................................................................................6Section2:DevelopmentandDescriptionoftheMethods.......................................................9Section3:ValidationofMaterialParticleMethod.................................................................13Section4:DevelopmentandDescriptionofMicrostructuresSimulations.............................15Section5:ParallelScalabilityAssessment.............................................................................23Section6:SummaryofMajorFindings..................................................................................24Section7:WorksCited..........................................................................................................25


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ExecutiveSummary The trabecular bone in the porcine skull is geometrically complex. It can be characterized experimentally, but requires many test configurations, loading rates, and samples to develop trusted constitutive models that fully characterize the complexity. Typically, Lagrangian finite element simulations are used in the bone modeling community to replicate experimental results for model validation and determination of material properties. In this approach, microCT images are used to develop anatomically accurate surfaces that are then volume meshed. While this modeling approach is valuable, there are some limitations. For example, with high-resolution micoCT data, traditional meshing techniques have proven to be rather difficult. Specifically with highly porous trabecular bone data, the complexity of the pore architecture is difficult replicate with a mesh. To overcome this challenge, the application of material point method (MPM) has been investigated for analyzing the material properties of trabecular bone. This meshless method requires a “particle mesh” that can easily be derived directly from the microCT data much easier than developing a finite element mesh. Preliminary results have focused on generating the stress-strain curves for quasi-static loading and comparing numerical predictions with experimental results, as well as verifying the MPM against the finite element method. Initial results seem promising and we have seen good comparison with experimental results. Parallel scalability of MPM is also considered for large-scale simulations.


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Acknowledgements I want to thank Dr. Kraft for providing me with great research resources and patient guidance. In this class, we got the chance to know about many advanced theories in nonlinear finite element method and practice them through Abaqus applications. These are valuable experiences for me. I also want to thank my collegues in the PSU Computational Biomechanics Group. Allison, Zac, Shruti and Ravi have given me a lot of help with my research as well as my life. I feel very happy and proud to be part of the group.


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ListofFigures Figure 1 a) THREE-DIMENTIONAL MODEL OF FULL PIG SKULL, b) CROSS-

SECTIONAL VIEW OF MICRO-CT SLICES WITH ENLARGED TRABACULAR BONE AREA. ......................................................................................................................... 6

Figure 2 3D SURFACE OF BONE SHOWING COMPLEXITY OF PORES ON SURFACE. ... 7Figure 3 a) SCAN IMAGE AND WITH NOISE, b) PARTICLES OF MPM MODEL, c)

PARTICLES IN DETAIL. ...................................................................................................... 9Table 1 SAMPLE DIMENSIONS AND PARTICLES ................................................................ 10Figure 4 INTERIOR GEOMETRY. a) SAMPLE 1, b) SAMPLE 2, c) SAMPLE 3, d) SAMPLE

4. EACH IMAGE SHOWS TWO PERPENDICULAR PLANES. ...................................... 10Figure 5 BOUNDARY CONDITIONS FOR MPM SIMULATION. .......................................... 11Figure 6 LOAD DIRECTIONS. a) NORMAL LOAD, b) TRANSVERSE LOAD. ................... 11Figure 7 a) BOTTOM STRESS vs. TIME, b) MPM-FEM COMPARISON. .............................. 14Figure 8 STRAIN RATE EFFECT. a) 2\% STRAIN, b) 20\% STRAIN. .................................... 16Figure 9 STRESS CONTOUR OF SAMPLE 4 UNDER COMPRESSION. a) 0\% STRAIN, b)

5\% STRAIN, c) 10\% STRAIN, d) 15\% STRAIN. ........................................................... 17Figure 10 MPM - EXPERIMENT COMPARISON OF SAMPLE 4. .......................................... 18Figure 11 a) DIFFERENT SAMPLES COMPARISON, b) DIFFERENT LOADS

COMPARISON. ................................................................................................................... 19Figure 12 STRESS CONTOUR OF SAMPLE 1 UNDER 15\% STRAIN. a) TRANSVERSE

LOAD, b) NORMAL LOAD. ............................................................................................... 20Figure 13 GEOMETRY OF SAMPLE 5 ...................................................................................... 21Figure 14 SIMULATIONS RESULTS OF SAMPLE 5. a) STRESS CONTOUR UNDER 3.8\%

STRAIN, b) MPM - EXPERIMENT COMPARISON. ........................................................ 22Figure 15 PARALLEL SCALABILITY OF MPM ...................................................................... 23


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Section1:BackgroundandProjectPlan Characterizing skull’s response to external loading requires an in-depth understanding of the material properties of trabecular bone which is an important composite of skull. As seen in Figure 1, trabecular bone is highly porous and porosity is highly dependent on the location in the skull. However, in skull modeling, trabecular bone is typically simplified as a homogeneous material with isotropic properties without considering the pores of trabecular bone [2-5]. Studying the nonlinearity and anisotropy of trabecular bone and its role in the overall response of skull may help gain insights into the failure mechanism of skull and human head.

Figure 1 a) THREE-DIMENTIONAL MODEL OF FULL PIG SKULL, b) CROSS-SECTIONAL VIEW OF MICRO-CT

SLICES WITH ENLARGED TRABACULAR BONE AREA.

The former work with FEM has shown that microstructure-level simulations gave good insight as to the response of trabecular bone under impact loading [6]. However, two major difficulties arise when applying FEM to trabecular bone from higher resolution scan data. First, using high-resolution microCT data to develop FE meshes is both difficult and costly. In order to develop accurate meshes that replicate the bone geometry well, elements must be small enough to capture pores as small as 1 micron. This is a time-consuming and involved procedure since it cannot be done automatically the bone structure is too complex for automatic meshing techniques. An example of this complex surface can be seen in Figure 2. Secondly, trabecular bone varies widely in bone geometries such as porosity and pore distribution throughout skull. Even if a single high quality mesh can be developed for one microstructure, developing additional microstructure models to capture the differences in porosity in different areas would be more time-consuming and unrealistic to complete.

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Figure 2 3D SURFACE OF BONE SHOWING COMPLEXITY OF PORES ON SURFACE.

To circumvent the issues with developing finite element meshes for complex geometries, meshless methods are desired. For example, the Material Point Method (MPM) is a meshless method in which Lagrangian point masses move though Eulerian background mesh. Since proposed in 1994 [7,8], this method has been proven to be very useful in dealing with complex-geometry and large-deformation problems. For example, human head impact simulation and modeling of response of woodpecker's head during pecking process show MPM’s advantage in dealing with CT scanning images from complex geometries [5,9]. Calculations of the full densification of foam microstructures by MPM have demonstrated consistency with various experimental results [10]. Applications in large strain engineering problems such as granular flow and plastic forming have been studied [11]. MPM also shows capability of simulating high explosive explosion problems and impact problems involving large deformation and multi-material interaction of different phases [12,13]. Compared with the traditional FEM, the MPM takes advantage of both Lagrangian method and Eulerian method. In this method, objects are discretized into a number of particles containing property information such as position, velocity and stress. Particles can be easily extracted from segmented CT data with little effort. A computational grid containing all the particles is created to receive and update the information stored in the particles. The grid deforms at each time step and is reset to its original configuration after updating information to particles. Therefore, the element distortion associated with the Lagrangian FEM is completely avoided. Also, unlike fully-Eularian codes, advection becomes less of a concern, as the grid moves with the particles for part of each time step. This project will outline the methods for developing MPM models and simulations to understand the bone behavior at the microstructure level. Firstly, MPM is verified by theoretical and FEM results with a solid cube model. Then simulations on high-resolution scanned microstructures with different compression loading are analyzed to study the material properties of trabecular bone. The simulations are also compared with experimental data to further verify the method as well as the computational models. Based on the microscale material behaviors, we aim to

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develop a macroscopic constitutive model that takes into account the geometric and material variability at the microstructural level and accurately models the fracture and failure of the bone.


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Section2:DevelopmentandDescriptionoftheMethods For this project, small scale samples from the skull of a six-month-old Gottingen mini pig were scanned at a resolution of 2.95 microns using a micro-computer tomography (microCT) scanner. Though it is possible to build MPM model on these high resolution images, the scan images are re-sampled with about 8 times resolution for current simulations for a balance of efficiency and accuracy. Computational models were developed from these re-sampled microCT scans. The greyscale scans were binarized to distinguish the bone from the rest of the image where white pixels represent bone, and black pixels represent everything else. Due to some noise in the images, the scans were post-processed slightly to remove floating bone that is not fully connected to the volume. These remaining volume points were then used to create the particle mesh as input for the MPM simulation. Since particles come directly from the microCT images, exact bone geometry is retained when developing the model (Figure3).

Figure 3 a) SCAN IMAGE AND WITH NOISE, b) PARTICLES OF MPM MODEL, c) PARTICLES IN DETAIL.

Four samples taken from different locations in the skull were modeled and analyzed in this paper. At the microscale of the bone, porosity is not constant – different locations in the skull have different levels and distributions of porosity. In order to characterize these differences and investigate how porosity plays a role in failure, each of the samples were take from a different area of the skull. The dimensions and total number of particles for each of the four samples are shown in Table 1. Figure 4 shows two cross sections (X and Y) of the geometries of the microstructures.

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Table 1 SAMPLE DIMENSIONS AND PARTICLES

Sample NO. 1 2 3 4 Dimensions 5.27*4.26*4.02mm 5.19*4.14*4.03mm 4.67*4.97*6.24 4.72*5.27*4.37 Particles 6.6 million 2.4 million 2.7 million 5.3 million

Figure 4 INTERIOR GEOMETRY. a) SAMPLE 1, b) SAMPLE 2, c) SAMPLE 3, d) SAMPLE 4. EACH IMAGE

SHOWS TWO PERPENDICULAR PLANES.

An open-source program, Uintah, was used for MPM simulation. Although many have shown the capability of the MPM code [14-19], this research compares Uintah simulation results to analytical and FEM results to further ensure the reliability of this code. Explicit MPM has been applied and various strain rates are simulated to determine the effects of loading rate on the overall response. The Generalized Interpolation Material Point Method (GIMP) [20] is employed

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in the current simulation and other more-advanced interpolation methods such as CPDI [21] and CPDI2 [22] will be considered for later simulations. Uniaxial compression simulations were run on one face of each microstructure model up to 15 percent Lagrangian strain. Example boundary conditions are shown in Figure 5. Samples 2 and 4 were loaded in the normal direction on the dorsal face of the sample that was closest to the skin while samples 1 and 3 were loaded in the transverse direction. Figure 6 shows the different loading direction.

Figure 5 BOUNDARY CONDITIONS FOR MPM SIMULATION.

Figure 6 LOAD DIRECTIONS. a) NORMAL LOAD, b) TRANSVERSE LOAD.

Compression Plate Microstructure

MPM Grid

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A linear elastic material model is used for the trabecular bone. The input material properties were taken based on comparison with the experimental results conducted by the United States Army Research Laboratory (ARL) on physical bone samples. The density is 2000 g/cm3, the Young’s Modulus is 1.5GPa and the Poisson’s Ratio of 0.22 was an estimate for trabecular bone [23,24].


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Section3:ValidationofMaterialParticleMethod A solid cube with dimensions of 5*5*5mm was used for this simulation. The MPM grid size was 100*100*100 and there were two particles in each direction each cell, namely there are 8 particles in each cell. The density is 2000 g/cm3, the Young’s Modulus is 0.75GPa and the Poisson’s Ratio is 0.24. Therefore, the theoretical stress wave velocity is 612.4 m/s. An obvious strain wave has been observed when compressing the cube with a constant velocity of 0.2m/s. From the plot of stress in Figure 7a, where stress values were taken from the bottom of the cube, we can see the time that the stress wave arrives is t=8.209-6s. Thus the simulated stress wave speed is 609.1 m/s. The simulated result is within 0.5\% error compared with the analytical result.


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Figure 7 a) BOTTOM STRESS vs. TIME, b) MPM-FEM COMPARISON.

A comparison with FEM simulations further shows the consistency between MPM and FEM when using LS-DYNA. With different grid sizes (25, 50 and 100) but the same number of particles in each cell (8 particles), all the MPM results compare well with FEM, as shown in Figure 7b.

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Section 4: Development and Description of MicrostructuresSimulations The former research using FEM has shown that there are minimal strain rate effects below a strain rate of 100 [6]. This conclusion is further verified by the MPM analysis. Figure 8a show the simulation results of Sample 4 with three different strain rates: 0.68s-1, 6.8s-1, 68s-1. For small strain (less than 2%), there is almost no difference in stress response between the three different strain rates. For relatively large strain (larger than 12%), there is a slight hardening caused by the dynamic loading (Figure 8b). Since this hardening is quite small, we still use the strain rate of 68s-1 (corresponding compressing velocity is 0.2m/s) for the following simulations to reduce computation cost.


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Figure 8 STRAIN RATE EFFECT. a) 2\% STRAIN, b) 20\% STRAIN.

From the stress contours under different compressing strain (Figure 9) of Sample 4, we can see that the stress magnitude is highly related to the local geometry. The area with more pores is subject to high stress while the area with less but larger pores has much smaller stress. Even at a small strain (5%), the stress on the highly porous area on the top of the microstructure is high. This indicates that in addition to bone volume fraction (BVF), we also need to consider the size and numbers of pores when assessing the failure of trabecular bone.

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Figure 9 STRESS CONTOUR OF SAMPLE 4 UNDER COMPRESSION. a) 0\% STRAIN, b) 5\% STRAIN, c) 10\%

STRAIN, d) 15\% STRAIN.

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Figure 10 MPM - EXPERIMENT COMPARISON OF SAMPLE 4.

Figure 10 shows the comparison of MPM with experimental data of Sample 4. Experimental data was adjusted to remove initial errors with machine compliance. After this adjustment, the MPM and experimental data are comparable as seen in Figure 10. From both experimental data and simulation results, we can see a considerable nonlinear relationship between stress and strain as strain increases. Comparing the results, there seems to be a greater nonlinear response in the experimental data, than in the simulation. This can be due to two things- material nonlinearities and geometric nonlinearities. Material nonlinearities were not considered in our model and thus cannot be captured in the response. Geometric nonlinearities in the response are due to the pores in the bone that cause the stress versus strain response to deviate from linear. Since some geometry details were eliminated from the model when the high-resolution CT data was resampled, some of these geometric nonlinearities may not be fully captured.

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Figure 11 a) DIFFERENT SAMPLES COMPARISON, b) DIFFERENT LOADS COMPARISON.

Figure 11a shows the simulation results of all the four microstructures. The input material properties and compression velocity are the same as that of Sample 4. Sample 2 and Sample 4 are loaded in normal direction and they have similar overall Young’s Modulus value. Sample 1 and Sample 3 are loaded in the transverse direction and they also show similar overall Young’s Modulus. However, Sample 1 and Sample 3 have larger overall Young’s Modulus than Sample 2 and Sample 3. This result verifies our assumption that trabecular bone is anisotropic and the


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failure response is dependent on the direction in which it is loaded. In the transverse direction, the trabecular bone is much stiffer than in the normal direction. However, the difference between different samples in normal and transverse direction varies with samples. Sample 2 has similar overall stiffness in normal and transverse direction while Sample 1 has almost 10 times larger stiffness in transvers direction than that in normal direction (Figure 11b). A comparison between transverse and normal load of Sample 1 (Figure 12) shows a very different stress distribution. For a transverse load, shown in Figure 12a, high stress appears near pore edges through out the compressing direction. For normal load (Figure 12b), high stress only appears around the top area in compressing direction and the overall stress is much smaller compared with that of the transverse direction. The causes for such difference still need to be further studied.

Figure 12 STRESS CONTOUR OF SAMPLE 1 UNDER 15\% STRAIN. a) TRANSVERSE LOAD, b) NORMAL LOAD.

The work presented above was completed for one set of microCT data. We have also been working on a second set of microCT data from a different Gottingen minipig specimen with the same resolution (2.95 microns). The goal of this subsection of work is to simulate physical experiments being conducted by ARL so as to validate generated models and procedures.

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Figure 13 GEOMETRY OF SAMPLE 5

An additional microstructure sample from this second skull dataset was modeled and compared with experimental data. This model, Sample 5, is 4.73*5.32*4.37mm and contains 5.3 million particles. The input material properties and compression velocity for the new sample are the same as that of Sample 4 and the compression direction is in the normal direction. From the geometries shown in Figure 13, the new skull bone has different geometrical structures compared to that of the other skull discussed above. While the top side (the dorsal face of the sample that was closest to the skin) still has many relatively small pores and the bottom side still has less pores with larger size, the porosity is smaller on the top side and larger on the bottom side compared to the former skull bone.


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Figure 14 SIMULATIONS RESULTS OF SAMPLE 5. a) STRESS CONTOUR UNDER 3.8\% STRAIN, b) MPM -

EXPERIMENT COMPARISON.

Figure 14 shows the simulation results compared with experimental data. The simulated results are very close to the experiment data at fist, but the engineering stress goes larger than the experimental results when strain is larger than 2%. The most possible explanation is that the skulls come from two different pigs, so the material properties probably are different. The real causes of the difference will be studied in the future work.

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Section5:ParallelScalabilityAssessment Scalability of parallel computing is an important aspect for numeric methods especially for big data analysis. With the given geometric complexity our trabecular bone samples roughly 2.4~6.6 million particles or elements are necessary to best represent the bone architecture and porosity. This number could further increase if larger samples or higher resolution scan data with more geometric details are desired for later simulations. Thus, computing scalability is essential in terms of time expense when problems scale up.

Figure 15 PARALLEL SCALABILITY OF MPM

A scalability test on MPM shows excellent parallel capability and great potential for larger problems. As shown in Figure 13, the log plots of simulation time versus number of cores are almost linear. The slope of the line is about -0.86=!"#!0.55 which means that for the same problem, the simulation time will be 0.55 times of the original time when the number of cores used for simulation doubles. Although the scalability slows down at 512 cores for the problem with 5.3 million particles and 75*75*75 grid, the linear scalability recovers immediately when the problem scales up.

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Section6:SummaryofMajorFindings Microscale simulations give a better understanding of the properties of trabacular bone. MPM has been verified to be a suitable tool in both accuracy and parallel scalability to model the complex geometries and relatively large deformations of trabacular bone. Two scan image sets from two different pig skulls have been simulated. Although both show good consistency with experimental results, further work must be completed to validate the models and understand the reason for certain results. Once complete, results from the microstructure simulations can be used for simulations at the full-head level.


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Section7:WorksCited [1] Martin, RB, B. D., and Sharkey, N., eds., 1998. Skeletal Tissue Mechanics. Springer-Verlag, New York. [2] Ruan, J., Khalil, T., and King, A. I., 1994. “Dynamic response of the human head to impact by three-dimensional finite element analysis”. Journal of biomechanical engineering, 116 (1), pp. 44–50. [3] Gilchrist, M. D., 2003. “Modelling and accident reconstruction of head impact injuries”. Key Engineering Materials, 245 , pp. 417–432. [4] Mao H. et al. “Development of a finite element human head model partially validated with thirty-five experimental cases”. J. Biomech. Eng. 135, 111002–111002-15 (2013). [5] Zhou S, Zhang X, Ma H (2013) “Numerical simulation of human head impact using the material point method”. International Journal of Computational Methods, 10.04 (2013): 1350014. [6] Ranslow*, A. N., Kraft, R. H., Shannon**, R., De Tomas-Medina**, P., Radovitsky, R., Jean, A., Hautefeuille, M. P., Fagan, B., Ziegler, K. A., Weerasooriya, T., Dileonardi, A. M., Gunnarsson, A., & Satapathy, S. "Microstructural Analysis Of Porcine Skull Bone Subjected To Impact Loading." Proceedings of the 2015 ASME International Mechanical Engineering Congress and Explosion (ASME IMECE). New York: American Society of Mechanical Engineers. [7] Sulsky D, Chen Z, Schreyer HL (1994) “A particle method for history-dependent materials”. Computer Methods in Applied Mechanics and Engineering 118: 179–196. [8] Sulsky D, Zhou S-J, Schreyer HL (1995) “Application of a particle-in-cell method to solid mechanics”. Computer Physics Communications 87: 236–252. [9] Liu Y, Qiu X, Zhang X, Yu TX (2015) “Response of Woodpecker's Head during Pecking Process Simulated by Material Point Method”. PLoS ONE 10(4): e0122677. [10] Bardenhagen, S. G., A. D. Brydon, and J. E. Guilkey. "Insight into the physics of foam densification via numerical simulation." Journal of the Mechanics and Physics of Solids 53.3 (2005): 597-617. [11] Więckowski, Zdzisław. "The material point method in large strain engineering problems." Computer methods in applied mechanics and engineering 193.39 (2004): 4417-4438. [12] S. Ma, X. Zhang, Y.P. Lian, X. Zhou, “Simulation of high explosive explosion using adaptive material point method”, Comput. Model. Eng. Sci. 39 (2) (2009) 101– 123. [13] Lian, Y. P., et al. "Numerical simulation of explosively driven metal by material point method." International Journal of Impact Engineering 38.4 (2011): 238-246. [14] J. Beckvermit, T. Harman, A. Bezdjian, Q. Meng, M. Berzins, C.A. Wight. “Parallel Multiscale Modeling of Transportation Accidents Involving Explosives,” XSEDE'14 Conference on High-Performance Computing, July, 2014. [15] A. Dubey, A. Almgren, John Bell, M. Berzins, S. Brandt, G. Bryan, P. Colella, D. Graves, M. Lijewski, F. Löffler, B. O’Shea, E. Schnetter, B. Van Straalen, K. Weide. “A survey of high level frameworks in block-structured adaptive mesh refinement packages,” Parallel and Distributed Computing, 2014. [16] A. Humphrey, Q. Meng, M. Berzins, D. Caminha B.de Oliveira, Z. Rakamaric, G. Gopalakrishnan. “Systematic Debugging Methods for Large-Scale HPC Computational Frameworks,” Computing in Science Engineering, Vol. 16, No. 3, pp. 48--56. May, 2014.


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[17] Q. Meng, M. Berzins. “Scalable large-scale fluid-structure interaction solvers in the Uintah framework via hybrid task-based parallelism algorithms,” Concurrency and Computation: Practice and Experience, Vol. 26, No. 7, pp. 1388--1407. May, 2014. [18] Qingyu Meng. “Large-Scale Distributed Runtime System for DAG-Based Computational Framework”, School of Computing, University of Utah, August, 2014. [19] M. Berzins, J. Schmidt, Q. Meng, A. Humphrey. “Past, Present, and Future Scalability of the Uintah Software,” Proceedings of the Blue Waters Extreme Scaling Workshop 2012, pp. Article No.: 6. 2013. [20] Bardenhagen, S. G., and E. M. Kober. "The generalized interpolation material point method." Computer Modeling in Engineering and Sciences 5.6 (2004): 477-496. [21] Sadeghirad, A., Rebecca M. Brannon, and J. Burghardt. "A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations." International Journal for Numerical Methods in Engineering 86.12 (2011): 1435-1456. [22] Sadeghirad, A., Rebecca M. Brannon, and J. E. Guilkey. "Second order convected particle domain interpolation (CPDI2) with enrichment for weak discontinuities at material interfaces." International Journal for Numerical Methods in Engineering 95.11 (2013): 928-952. [23] Dalstra, M., Huiskes, R., Odgaard, A., and Van Erning, L., 1993. “Mechanical and textural properties of pelvic trabecular bone”. Journal of biomechanics, 26 (4), pp. 523–535. [24] Horgan, T., and Gilchrist, M. D., 2003. “The creation of three-dimensional finite element models for simulating head impact biomechanics”. International Journal of Crashworthiness, 8 (4), pp. 353–366.

me563 project final - psu compbio

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