Urban Habitat Constructions under Catastrophic Events (Proceedings) – Mazzolani (Ed).© 2010 Taylor & Francis Group, London, ISBN 978-0-415-60685-1

A feasibility study on modeling blast loading using ALE formulation

, M. Balcerzak & J. WojciechowskiWarsaw University of Technology, Warsaw, Poland

ABSTRACT: The paper presents a comparison of numerical data for computer simulations of blast loading.Based on the published experimental data, a benchmark problem was selected, where the pressure loadingsubjected to a rigid steel plate and produced by near field hemispherical charges, is considered. The comparisonis done in terms of peaks of reflected pressure (overpressure), reflected specific impulses, and time histories ofreflected pressure. The numerical study was conducted using mainly three dimensional Arbitrary LagrangianEulerian (ALE) formulation implemented in the commercial code LS-DYNA®. Several other modeling optionsand strategies were also considered and validated through comparison with the available experimental data. Thesensitivity study on mesh resolution for ALE meshes was also considered. Although several discrepancies wereindicated, the premature validation study shows big potential for this type of numerical modeling.

1 INTRODUCTION

The experimental and numerical studies on struc-tures subjected to blast loads show complexity of theproblem and many challenges facing this researcharea, Bulson (1997). Among the most fundamentalobjectives, there can be mentioned three: a reliableprediction of loads imposed on structures by explo-sives, correct representation of material behavior, andglobal analysis of large scale structures.

This paper reports a preliminary feasibility studyon approaches for numerical modeling of blast loads,implemented in the commercial program LS-DYNA®

(Hallquist 2009). Four approaches have been con-sidered including: explicit blast wave representationusing fluid-structure interaction with 3D and 2D multi-material arbitrary Lagrangian-Eulerian (ALE) formu-lations, direct application of empirical explosive blastloads on structures, and the most recent, combinedmethod, in which direct empirical loading is subjectedto the ALE domain. Each of these approaches has itsown advantages and limitations, although the last oneseems to be the most universal. Numerical models offluid-structure interaction with 3D ALE formulationare able to capture complex interaction of reflectedand dispersed blast waves but are computationallyvery demanding and require very dense meshes. TheALE elements must be small enough to capture thenearly discontinuous shock front of the blast wave.The 2D approach is computationally much more effi-cient but is only applicable to simple axisymmetricproblems. The application of direct empirical blastloads is the most effective method but it is limited tofree-air cases where there are no interferencing objectswhich can cause reflecting or focusing of blast waves.To overcome this limitations a new method has beendeveloped recently in LS-DYNA® which combines

direct empirical load with fluid-structure interactionusing ALE 3D domain (Slavik 2009).

2 SELECTED BENCHMARK PROBLEM

At this stage of the reported feasibility study thenumerical results are compared with published exper-imental data for a selected problem investigated byPope & Tyas (2002). In a series of experiments, 78 gPE4 hemispherical explosive charges were used toload a target plate at short stand-offs, varying from400 to 1000 mm. With special care taken to producerepeatable testing conditions, Pope & Tyas obtainedhighly reliable pressure time histories recorded bypressure transducers placed in the target semi-rigidplate. A typical test setup is presented schematically inFigure 1. Comparison is done in terms of arrival time,peaks of reflected pressure (overpressure), reflectedspecific impulses for the time histories of reflectedpressure at the central point of the target plate.

Obtained experimental results are in good cor-relation with approximations given by well-knownFriedlander function (Slavik 2009), with the maximumreflected pressure decreasing and arrival times increas-ing for increased stand-offs. It was also confirmedthat at the near field stand-offs the charge shape isan important factor and can result in different loadingmagnitudes for the same charge masses.

3 FLUID-STRUCTURE INTERACTIONUSING ALE FORMULATION

3.1 Arbitrary Lagrangian-Eulerian formulation

The ALE methods enable to conduct calculations forcoexisting Lagrangian and Eulerian meshes and inthis way to reproduce interaction between fluid and a

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Figure 1. Test setup for a series of experiments used byPope & Tyas (2002) to produce time histories of reflectedoverpressure.

deformable structure.The pure Lagrangian descriptionmakes it easy to track interfaces and to apply bound-ary conditions. The ALE mesh, created to representthe fluid domain, is usually undistorted (can be auto-matically expanded) and might be fixed (as in typicalEulerian formulation) or can move according to theprescribed, arbitrary conditions, Hallquist (2009). AnALE mesh can be built of multimaterial elements orelements filled with a fluid and void. Interfaces andboundary conditions are difficult to track using thisapproach; however, mesh distortion is not a problembecause the mesh is smoothed to its original shapeand dimensions after each calculation cycle. In themodel, fluid flows through the undistorted mesh and,due to coupling algorithms, interacts with a structurerepresented by a Lagrangian mesh or rigid boundaryconditions. The Lagrangian mesh is independent andusually is immersed in the ALE regular mesh whichdefines the space where the fluid motion is traced.

The ALE and Eulerian methods implemented inexplicit FE codes such as LS-DYNA® are basedon the classical Lagrangian formulation (Hallquist2009). The solution algorithm is incremental con-sisting of finite element explicit (Lagrangian) timesteps followed by so called advection steps. For pureLagrangian approach, where a mesh moves with thefluid, extensive element distortions are likely to occurwith the mesh degeneration leading to small or neg-ative volumes. To avoid numerical problems due tolarge distortions, the finite element mesh is smoothedwhen deformations become too large during the com-putation. The ALE formulation enables cyclic andautomatic rezoning, based on the deformed boundariesbetween Lagrangian cycles. The advection step carriesout an incremental remapping followed by the trans-portation (advection) of nodal and element variables,from the distorted to the new mesh. The computationalcost of an advection step is typically two to five timesthe cost of the Lagrangian time step (Hallquist 2009).

For blast analysis the ALE domain is filled withair modeled as ideal gas with the linear polynomialequation of state (EOS). The charge is modeled usingspecial purpose material model named High ExplosiveBurn characterized by the Jones-Wilkins-Lee EOS

Figure 2. Definition of hemispherical charge usingoption “initial volume fraction geometry” for 2D mesh2.5 × 2.5 mm.

(Hallquist 2009). In this way the explosive as well asthe air are explicitly modeled allowing for tracing theblast wave propagation through the ALE air domainand interaction with the Lagrangian structure throughfluid-structure interaction.

More detailed description of this novel formulationcan be found in Donea (1980) and Hallquist (2009).

3.2 Axisymmetric 2D models

An axisymmetric model is built of regular squareelements with the symmetry axis set default alongthe y-axis of the model. The shape of the explo-sive material is cut out from the domain defined bymulti-material ALE elements using “initial volumefraction geometry” option, as shown in Figure 2. Thetarget plate is represented by boundary conditions con-straining y-translation of the ALE nodes at the spec-ified distance from the explosive material. All otherboundaries have default, nonreflecting conditions.Thecalculations were carried out for meshes with dif-ferent element sizes: 5 × 5 mm, 2.5 × 2.5 mm, and1.25 × 1.25 mm to verify the effect of mesh densityon the results.

Figure 3 shows contours of pressure showing prop-agation of blast wave for 500 m stand-off. Figure 4presents time histories of reflected overpressure forall considered stand-offs. It can be noticed that thearrival time of the blast waves is the same as for theexperiments but the shape of the function differs fromthe typical Friedlander description. The peak reflectedoverpressures, calculated and from the experimentsare compared in Figure 5. As expected the values arehigher for smaller element size, but for the small-est considered element size of 1.25 mm the solveroverestimates the results. Additionally, for the finermeshes some disturbances in reflected overpressurehistories appear. Figure 6 presents comparison of cal-culated specific impulses with the experiment. Thesame as in the case of reflected overpressure, resultsare overestimated for the finest mesh in comparisonto the experiment and other mesh densities indicatingcomputational problems of the solver.

3.3 ALE 3D model

Several modeling issues have to be considered fordirect simulation of fluid-structure interaction with 3DALE formulation. First, to reproduce properly the blastwave propagation, the air domain must be represented

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Figure 3. Contours of pressure showing propagation of blastwave for 500 m stand-off and element size 2.5 mm.

Figure 4. Time histories of reflected overpressure for dif-ferent stand-offs and element size 2.5 mm.

Figure 5. Comparison of peak reflected overpressure forexperiment and 2D ALE with element sizes 1.25, 2.5, and5 mm.

Figure 6. Comparison of specific impulses for experimentand 3D ALE models with element sizes 1.25, 2.5, and 5 mm.

by a regular ALE mesh built of hexahedral elementspreferably with the aspect ratio of unity. To includecharges, two options can be applied.

The material model for explosive charge can beassociated with selected finite elements of the ALEmesh but with a rectilinear mesh this option is appli-cable only to cubical charges. For charges with com-plex shapes, an option called “initial volume fractiongeometry” (Hallquist 2009) can be applied. This is a

Figure 7. Views of a quarter of the hemispherical chargedefined using volume-filling command “initial volume frac-tion geometry”.

Figure 8. Snapshots of iso-surfaces of pressure for 400 mmstand-off modeled with 3D ALE mesh of 5 mm.

Figure 9. Time histories of reflected overpressure for ele-ment size 5 mm and different stand-offs.

volume-filling command used for defining the volumefractions of the multi-material ALE elements whichcan be occupied by a selected material. Figure 7 showsapplication of this option for generation of one quarterof the hemispherical charge cut out with the symmetryplanes (compare Fig. 2).

To reduce number of finite elements, only one quar-ter of the volume is represented for the cubical spacewhere the blast wave is traced. The modeled spaceis bounded by two symmetry planes with boundaryconditions constraining displacements in the perpen-dicular direction. The target plate is also representedby constrained perpendicular displacements of theselected nodes. On the other external surfaces thedefault nonreflecting boundary conditions are set.The 3D ALE model calculations were carried out forthe finite element sizes 5 and 10 mm.

Figure 8 shows snapshots of iso-surfaces of pres-sure for 400 mm stand-off modeled with 3DALE mesh

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Figure 10. Comparison of peak reflected overpressure forexperiment and 3D ALE models with element sizes 5 and10 mm.

Figure 11. Comparison of specific impulses for experimentand 3D ALE models with element sizes 5 and 10 mm.

of 5 mm. Figure 9 presents example time histories ofthe reflected overpressure for element size 5 mm. Thesame as in the case of the ALE 2D model arrival timeof blast wave is practically the same as for the experi-ment. Figures 10 & 11 present comparisons of the peakreflected overpressures and of the specific impulses,respectively. Although the results are underestimatedin the reference to the experimental data, it can bepredicted that further refinement of the mesh wouldimprove the results, because in contradiction to the 2DALE model (section 3.2) the refinement itself does notcause disturbances of the time histories for reflectedoverpressure.

4 EMPIRICAL BLAST LOADS

In this approach, air blast pressure is computedwith empirical blast equations and directly appliedto Lagrangian elements of the structure. The methodtakes an advantage of well calibrated empirical equa-tions derived using a compilation of results from largenumber of explosive air blast experiments (Slavik2009). Without modeling explicitly the air betweenthe explosive and the structure this approach is com-putationally very effective, especially for global anal-yses when large stand-off distances and large scalestructures are considered. The disadvantage of thisapproach is that it cannot capture interaction of theblast wave with any objects in front of the analyzedstructure.

Table 1. Comparison of peak reflected pressure and arrivaltime for experiment and empirical blast loads.

Stand-off 400 mm 1000 mm

Peak pressure [MPa]experiment 12.4 0.7calculation 78 g spherical charge:elem. size 25 × 25 mm 5.953 0.132elem. size 10 × 10 mm 5.597 0.408elem. size 5 × 5 mm 1.073 0.413calculation 78 g hemi-spherical charge:elem. size 10 × 10 mm 9.684 0.648calculation 1.37∗78 g spherical charge:elem. size 5 × 5 mm 7.804 0.552calculation 1.37∗78 g hemi-spherical charge:elem. size 5 × 5 mm 12.512 0.900

Arrival times of shock wave [ms]experiment 1.52 8.58calculation 78 g spherical charge:elem. size 25 × 25 mm 1.88 10.03elem. size 10 × 10 mm 1.99 10.03elem. size 5 × 5 mm 1.99 10.03calculation 78 g hemi-spherical charge:elem. size 10 × 10 mm 1.72 9.66calculation 1.37∗78 g spherical charge:elem. size 5 × 5 mm 1.83 10.14calculation 1.37∗78 g hemi-spherical charge:elem. size 5 × 5 mm 1.61 8.85

To determine the blast loading a user needs toselect among several cases and define the chargeweight and its position relative to the structure. Twodifferent stand-offs were considered – 400 mm and1000 mm. For both cases the target square steel plateshown in Figure 1, was modeled with the fully inte-grated shell elements with two different mesh densitiescorresponding to element sizes of 25 × 25 mm and5 × 5 mm. All nodes were fully constrained to imi-tate the rigidity of the plate. The loading was appliedusing “load blast enhanced” option defining the chargemass of equivalent to TNT, its distance from the targetand conversion factors (in this study millimeters, tonsand seconds were used). This option allows model-ing the action of shock wave for different chargeshapes. Unfortunately, the considered benchmark con-figuration, with the hemispherical charge (shock wavewithout reflection and reinforcement from ground), iscurrently not available in the program LS-DYNA®. Forthe “load blast enhanced” option one can choose eitherspherical charge without reflection of shock wave,or hemispherical charge situated near to the ground.Shape of the charge in both cases has some impact onthe results, as it is shown further. Comparison of theexperimental (Pope & Tyas 2002) and the calculatedresults for different cases considered is presented inTable 1.

Example time histories of the reflected pressurecalculated for the spherical charge, 400 mm stand-offand three mesh densities are presented in Figure 12.Figure 12 shows that for the shell element sizes 5 mm

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Figure 12. Time histories of reflected pressure for spheri-cal charge, 400 mm stand-off, and for three different meshdensities.

Figure 13. Time histories of pressure for spherical andhemispherical charges and 400 mm stand-off.

Figure 14. Contours of pressure applied to the target palateand generated by option “load blast enhanced”.

and 10 mm the estimated blast loading applied to thetarget plate in terms of the overpressure is almost iden-tical.The differences between the results calculated forthe same shell element sizes 10 × 10 mm and for thehemispherical and the spherical charges and 400 mmstand-off, are presented in Figure 13. The distributionof the blast loading generated by the option “load blastenhanced” and applied to the target palate is shown inFigure 14 as contours of pressure.

The peak reflected overpressure generated in themiddle point of the target plate for the stand-off

400 mm was calculated as equal to 5.95 MPa in thecase of hemispherical charge and 9.68 MPa for thespherical charge. For the stand-off 1000 mm the val-ues are accordingly 0.41 MPa and 0.65 MPa. Thecorresponding experimental data gives 12.4 MPa and0.7 MPa for 400 mm and 1000 mm stand-offs, respec-tively. The numerical results are affected by the factthat the empirical formula “Load blast enhanced” isset for TNT, and in the experiment the PE4 chargewas used. These materials are quite similar, howeverdetonation velocity of PE4 is higher than for TNT(Bulson 1997). Therefore the time, when the shockwave reaches the plate, may be slightly different –in the experiment for the 400 mm stand-off the timeequals to about 1,54 ms, whereas in the calculationsit is 2 ms (for 1000 mm stand-off it is accordingly8,56 ms and 10,03 ms). Also the pressure on the platefor the same mass ofTNT and PE4 is smaller in the caseof TNT, however it is also predictable and accordingto Weckert & Anderson (2006) for the peak pressurethe equivalence ratio between TNT and PE4 is 1.37.If the mass of the charge is multiplied by 1.37 thepeak pressure reaches 12.51 MPa for 400 mm stand-off and 0.90 MPa (0.55 MPa for the spherical charge)for 1000 mm stand-off (see Table 1). The effectiveimpulses obtained for the mass of charge multipliedby 1.37 are also presented in Table 1. For 400 mmstand-off the effective impulse is equal to 0.54 kPa·msfor the hemispherical charge and 0.33 kPa·ms forthe spherical charge, while the experimental valueequals to 0.38 kPa·ms. For the 1000 mm stand-off thereare accordingly 0.16 kPa·ms and 0.10 kPa·ms, withthe experimental value equal to 0.13 kPa·ms. Theseresults seem to be quite reasonable comparing to theexperiment.

5 COMBINED METHOD

In the new, combined method, only the air immedi-ately surrounding the Lagrangian structure needs tobe explicitly modeled with the ALE mesh. The blastpressure calculated using empirical blast equations(section 4) is directly applied to the most outer faceof the ALE air domain. In this way effects such asfocusing and shadowing, caused by interfering objectsimmersed in the ALE mesh, can be captured. The newmethod is supposed to accommodate a good balancebetween accuracy and computational efficiency.

In the case described below the same test setupas presented in Figure 1 was applied with a 78 gspherical TNT charge at stand-off of 400 mm. Thecalculations were conducted for different ALE meshdensities: 20 × 20 × 20 mm, 10 × 10 × 10 mm and5 × 5 × 5 mm. Figure 15 shows the FE model with theambient layer positioned 200 mm from explosive andthe ALE mesh for air. The rest of the air between thecharge and the ALE mesh and the charge itself are notexplicitly modeled. In this way the model has substan-tially reduced number of elements. The empirical blastload is applied directly to the outer face of the ALEmesh through the ambient layer.

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Figure 15. FE model for combined method.

Table 2. Comparison of peak reflected pressure and arrivaltime for experiment and combined method.

Stand-off 400 mm

Peak pressure [MPa]experiment 12.4elem. size 20 mm 1.58elem. size 10 mm 2.59elem. size 5 mm 3.52

Arrival time of shock wave [ms]experiment 1.52elem. size 20 mm 1.55elem. size 10 mm 1.68elem. size 5 mm 1.80

Figure 16. Iso-surfaces of pressure for the stand-off of400 mm modeled with combined method.

The comparison of peak reflected pressure andarrival time for experiment and combined method ispresented in Table 2. Figure 16 presents snapshotsof pressure iso-surfaces showing propagation of theshock wave through air modeled as the ALE domain.The time histories of reflected pressure calculatedfor different mesh sizes are compared in Figure 17.

Figure 17. Time histories of reflected pressure for spheri-cal charge, 400 mm stand-off, and for three different meshdensities.

Although the increased mesh density improves qualita-tively the results the peak values are still substantiallyunderestimated. The early stage results suggest thatthe finer ALE meshes are required to get compa-rable peak pressures for short stand-offs (comparesections 3.2 & 3.3).

6 CONCLUSIONS

For all computational approaches with the ALE for-mulation the results appeared to be highly sensitiveto mesh density especially for the peak values of thereflected overpressure. The numerical estimation isbetter for longer stand-offs and is more precise forthe effective impulses than for the peak values of pres-sures. The effective impulse is considered as the mostimportant damage causing factor of the blast.Althoughseveral discrepancies were indicated, the prematurevalidation study shows big potential for this type ofnumerical modeling.

REFERENCES

Bulson, P.S. (1997) Explosive Loading of Engineering Struc-tures, E & FN SPON.

Donea, J. 1980. Advanced Structural Dynamics, AppliedScience Publishers LTD, London UK.

Hallquist, J.O. 2009. LS-DYNA Keyword Manual,Version 971.Livermore: Livermore Software Technology Corporation.

Pope, D.J. & Tyas, A. 2002. Use of hydrocode modellingtechniques to predict loading parameters from free airhemispherical explosive charges. In 1stAsia-Pacific Con-ference on Protection of Structures Against Hazards,Singapore, November 2002.

Slavik, T.P. 2009. A Coupling of Empirical Explosive BlastLoads to ALE Air Domains in LS-DYNA®, 7th EuropeanLS-DYNA Conference 2009.

Wecker, S. & Anderson, Ch. 2006. A Preliminary Com-parison Between TNT and PE4 Landmines , WeaponsSystem Divisoin, DSTO Defence Science and TechnologyOrganisation

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