Army Research Laboratory 

Aberdeen Proving Ground, MD 2 1005-5069 

ARL-TR-2303 Sedember 2000 

Practical Aspects of Numerical 

Simulations of Dynamic Events: 

Effects of Meshing 

Jonas A. Zukas 

Computational Mechanics Consultants, Inc. 

Daniel R. Scheffler 

Weapons and Materials Research Directorate, ARL 

Approval for public release; distribution is unlimited. 
Abstract 

The use of finite-difference and finite-element computer codes to solve problems involving 

fast, transient loading is commonplace. A large number of commercial codes exist and are 

applied to problems ranging from fairly low to extremely high damage levels (e.g., design of 

containment structures to mitigate effects of industrial accidents; protection of buildings and 

people from blast and impact loading; foreign-object impact damage; and design of space 

structures to withstand impacts of small particles moving at hypervelocity, a case where the 

pressures generated exceed material strength by an order of magnitude). But, what happens if 

code predictions do not correspond with reality? This report discusses various factors related 

to the computational mesh that can lead to disagreement between computations and experience. 

Subsequent reports will focus on problems associated with contact surfaces and material 

transport algorithms, constitutive models, and the use of material data at strain rates 

inappropriate to the problem. It is limited to problems involving fast, transient loading, which 

can be addressed by commercial finite-difference and finite-element computer codes. 

This report has been accepted for publication in a future volume of the International Journal 

of Impact Engineering. 

ii 
Acknowledgments 

The authors would like to thank Dr. Steven B. Segletes, who served as technical reviewer for the 

original paper submitted to the International Journal of Impact Engineering and 

Mr. Stephen Schraml, who served as the technical reviewer for this report version of it. Their 

thorough reviews and comments helped improve the paper and technical report. 

. . . 

111 
iv 
Table of Contents 

p&e 

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111 

vii 

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Vll 

1 

3 

7 

9 

10 

11 

15 

16 

24 

29 

31 

35 

43 

1. 

2. 

3. 

4. 

5. 

5.1 

5.2 

5.3 

5.4 

6. 

7. 

Acknowledgments ................................... ................ 

List of Figures ................... .................................... 

ListofTables ....................................................... 

Introduction ............... ........................ ................. 

Difficulties Inherent in NumeriCal Approximations ........................ 

Idealization ............................ 

............................. 

The Human Factor .......................... ......................... 

Problems Related to Computational Meshes ............ .................. 

Element Aspect Ratio ................. ............................. 

ElementArrangement ...... ........................................ 

Uniform and Variable Meshes 

........................................ 

Abrupt Changes in Meshes ........ .................................. 

Summary ............. .......................... .............. ...... 

References ................ ...... ......... ...... ..................... 

Distribution List ............................... 

......... ............. 

ReportDocumentationPage.. ................................... ...... 

V 
T.NT~~N-IT~NAUY LEFT BLANK. 
List of Figures 

Figure 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

Variation of Computational Results With User Experience .................... 

Element Orientations in the EPIC-2 Code .................................. 

Comparison of Elementary and Skalak Solutions for Bar Impact (Skalak Solution Is 

DrawnFreehand) ..................................................... 

Signal Filtration With Mesh Aspect Ratio ................................. 

Effects of Element Arrangement on Plastic Strain Distribution ................. 

Grid Effects on Long-Rod Penetration .................................... 

Gradual vs. Abrupt Grid Changes ........................................ 

Fragment Size Distribution vs. Grid Size .................................. 

Grid and Contact Surface Effects for Oblique Penetration ..................... 

Abrupt Grid Change Effects on Penetration ................................ 

Residual Penetrator and Hole Size in the Presence of an Abrupt Grid Change ...... 

(a) Abrupt Grid Change, (b) Gradual Change in Element Size, and (c) Uniform 

Meshing ............................................................ 

4 

12 

13 

14 

17 

18 

19 

22 

23 

26 

27 

28 

List of Tables 

Table m 

1. Dynamic Situations . ..*..........*.............................,....., 2 

vii 
. 

. . . 

Vlll 
.__ 

1. Introduction 

c This report focuses on the numerical simulation of problems in mechanics involving fast, 

transient loading. For practical purposes, these can be divided into problems involving structural 

dynamics and wave propagation (Table 1). There is no clear demarcation between these two areas. 

The labels are misleading since both types of problems involve wave propagation. Nonetheless, 

these designations have caught on in the literature. These two labels deal with the behavior of inert 

materials that are subjected to intense impulsive (distributed over a surface, such as air blast, over 

a long time [milliseconds to seconds]) or impact (applied to a single point or a very small area over 

a very short [nanoseconds to microseconds] time span) loading. 

There is also a large class of energetic materials that reacts quite differently when excited. 

Energetic materials are not discussed in this report. For more information on this topic see Zukas 

and Walters [ 11, Cooper [2], Cooper and Kurowski [3], Cheret [4], Blazynski [5], Fickett and 

Davis (61, and Mader [7, 81. 

Most of the work done in the area of fast, transient loading is experimental in nature. This is 

due either to complexities of geometry or the nonlinearity of material behavior or both. Closed-form 

analytical solutions are generally rare and apply only to some small subset of the overall problem. 

Numerical solutions, in the form of finite-difference and finite-element codes, have been 

successfully used in the past. In particular, the combination of experiments, numerical solutions, and 

dynamic material characterization has been shown to be very effective in reducing both manpower 

requirements and cost [9]. However, the computer codes available for dynamic analyses are quite 

complex. Considerable experience with both the codes and the physical problems they are intended 

to solve is vital. Also critical is the determination of material constants for the various constitutive 

models available at strain rates appropriate to the problem. 
Table 1. Dynamic Situations 

STRUCTURAL DYNAMICS 

l Blast/Shock Loading of Structures 

. Underwater Explosions 

l Fluid-Structure Interactions 

l Mechanical System Dynamics 

- machinery & mechanisms 

- agricultural, construction, off-highway equipment 

- turbomachinery systems 

- containment structures 

- vehicular collisions 

- aeronautical/aerospace systems 

l Biodynamic Systems 

l Plate and Shell Structures 

l Nonperforating Impacts 

l Rotating Machinery 

l Metal Forming 

WAVE PROPAGATION 

l Lunar/Planetary Impact 

l Explosive Welding, Forming, Compaction 

l Shock Consolidation/Shock Synthesis 

l Chemical Energy Penetrators 

- explosively formed penetrators (EFP) 

- shaped charge jets 

l Kinetic Energy Penetration 

- fragments 

- long rods 

- bombs 
c 

Figure 1 shows the dramatic effect that experience can have on computed results. The figure, 

provided by Dr. Paul Senseny of the former Defense Special Weapons Agency (now called Defense 

Threat Reduction Agency), shows the results obtained by four different users of the DYNA code for 

a problem involving airblast loading of a silo door. Each of the code users worked independently. 

Note that four people with four different backgrounds produced four distinctly different results with 

the same code in solving the same problem. 

Why this emphasis on experience? To be sure, experience is not a sufficient factor to guarantee 

accurate computations. Indeed, all four code users in the previous problem could have been 

experienced in some sense. But experience is an important factor. There exist many cases in 

computational continuum dynamics where tradeoffs must be made to achieve reasonable results in 

a reasonable time at finite costs. Szabo and Actis [lo] point out the importance of timely solutions 

in industry. Quoting from an engineer’s experience at a major industrial laboratory, they point out 

that “...the stress engineer may not be able to guide the design because by the time he has generated 

his several thousand degree-of-freedom finite-element model, prototypes are already being made.” 

Anyone can perform finely resolved one-dimensional (1-D) calculations almost by rote. 

Two-dimensional calculations (2-D), though now largely performed on workstations and personal 

computers, require a keen knowledge of the problem and numerical simulation methods. This holds 

even more for successful three-dimensional (3-D) simulations that, with all their compromises, will 

still be expensive and require significant computer resources. 

2. Difficulties Inherent in Numerical Approximations 

When a user acquires a code, he/she receives a package in which certain decisions have been 

made by the developer. The user has no control over some of the approximations that come as part 

of the package but must be aware of them, as they will affect all solutions that the user will generate 

with the package. This section briefly describes these errors and cites sources in the 

finite-difference/finite-element literature where extensive discussions can be found. These are 

problems inherent in transforming a physical problem into a discrete model and solving it on 

3 
The Same Code Will Produce Different Answers 

In The Hands of.Different Users 

0 Em loo0 

Time, psec 

DYNA was used in all calculations 

Figure 1. Variation of Computational Results With User Experience. 

4 
computers with limited precision through the medium of finite differences and finite elements. Each 

difference scheme and each element have unique characteristics that can affect the numerical 

solution. For example, most implicit schemes for advancing the solution in time have a certain 

amount of viscosity built into them over which the user generally has no control. Yet, the user must 

be aware of this characteristic when he/she sees his/her numerical solution drift out of phase with 

an experimental or analytical result. Certain finite elements are prone to locking (becoming 

excessively stiff) when the constitutive relationship is evaluated at element Gauss points. This can 

be overcome by evaluating the constitutive equations at one point (usually the element centroid). 

This procedure is called “underintegration.” It eliminates locking but gives rise to spurious 

deformation modes known as “hourglassing.” A user, unaware of this, might interpret such 

numerical noise as a physical response. Explicit integration schemes are only conditionally stable. 

Using stability fractions built into explicit hydrocodes without consideration of the problem being 

solved can lead to unstable solutions that can degrade over hundreds of cycles and could be 

interpreted by novice code users as a physical response of the system being modeled. 

The errors inherent in fmite-element modeling are nicely discussed by Utku and Melosh [ 111, 

as are guidelines for mesh preparation by Melosh and Utku [ 121. A superb work on practical aspects 

for static and dynamic finite-element analyses is the book by Meyer [ 131. Extremely valuable 

insights into finite-element design and its practical application are to be found in MacNeal’s 

book [ 141. Morris and Vignjevic [ 151 review error control and error bounding methods for finite 

elements. They also present a method for error control in the idealization phase of a full 

finite-element analysis. These, together with personal experience, are the main references for the 

material in this section. 

An analyst begins by examining some physical problem that needs to be modeled numerically. 

This may be necessary because it is too expensive, too difficult, or just plain impossible (as in 

nuclear testing) to perform parametric experiments. It might be because the information needed does 

not lend itself to direct measurement. It is often the case [9, 161 that the combined use of 

experiments and calculations, including some material characterization at high strain rates, produces 

more information in less time and at less cost than reliance on experiments or calculations alone. 

5 
At first glance, a problem may appear to be extremely complex. Pressure vessels come with various 

cutouts, end caps that can range from circular plates through hemispherical shells, pipes coming in 

and out, some irregular sections, weld points, and a countable infinity of bolts and assorted 

restraining devices. Face it, real structures are messy to model numerically. They can be mounted 

to floors and walls and receive and transmit loads to these through a variety of mounts (shock 

isolation is popular in Japan and other earthquake-prone areas). The space shuttle has a nice, clean 

shape, but most space structures tend to show greater similarity to the highly irregular surfaces of 

the space cruisers in Star Wars than to the shuttle. Thus, high-velocity impact-generated debris and 

ricocheting projectiles can interact with critical, externally mounted equipment such as antennas and 

solar panels. 

Three-dimensional impact problems are the rule rather than the exception. Uniform grid 

resolution is generally not possible in practical problems. Some gradation of the mesh is required. 

If this is not done with great care, spurious signals and assorted numerical artifacts arise in the 

calculation, leading to instabilities or a masking of the desired results. Even coarse calculations can 

require in excess of 40 central processing unit (CPU) hours on modern computers. If sufficient 

memory is not available to run the problem “in-core,” then extensive buffering between main 

memory and mass storage is required, further increasing turnaround time and cost. For sufficiently 

large problems on a small central memory machine, it is possible to approach situations where the 

bulk of CPU time is spent on input-output operations and only a small fraction is spent in advancing 

the solution, rendering the computation uneconomical. 

Does this mean that 3-D calculations cannot be done today? Not at all. Quite a few practical 

problems have been successfully addressed with 3-D codes. However, compromises are required, 

and these, in turn, require a keen understanding of the physical problem, the effects of discretization 

on that problem, and the effects that numerical artifacts (such as uneven resolution in different 

coordinate directions, mixing of implicit and explicit integration schemes or explicit-explicit 

partitions, choice of mesh or element type, effects of sliding surfaces or interfaces, and the use of 

various viscosities to stabilize computations) have on the solution. 

6 
All the details and materials in a real structure usually cannot be accounted for in a numerical 

simulation. Hence, the analyst must now make an analytically tractable model without sacrificing 

the essential elements that make up the response of the physical structure. The fust order of business 

is to simplify the physical system by taking all essential geometric and material features that govern 

its response into account. Some simplification of geometry and lumping of masses is inevitable. 

Depending on the situation, one may be able to employ specialized mathematical models such as 

beam, plate, or shell theory (or a combination of these). Prelinrinary analyses need to be made to 

determine whether large strains and rotations constitute a part of the response or whether a linear 

analysis will suffice. A number of uncertainties are introduced in this process, some of which cannot 

be quantified. Other uncertainties are due to variabilities in material properties, loading, fabrication, 

and other factors. For example, rolled homogeneous armor (RHA) steel is used extensively in 

military construction. It can safely be said that RHA is rolled. It is also used as armor. However, 

the military specifications that govern the production of RI-IA have wide tolerances so that it is 

anything but homogeneous. Material properties (primarily hardness) are known to vary by as much 

as 10% within a lot of RHA and up to 30% from lot to lot. This makes single tests (the famous 

“one-shot statistics”) useless and correlation between numerical ,results and experiments unlikely 

unless a statistically meaningful number of tests have been done. Simple go/no-go ballistic tests can 

cost upward of $2,000 each. Instrumented field tests can run from $10,000 to $100,000 each. As 

a rule, then, a statistically significant data set is almost never available. 

3. Idealization 

The ultimate goal of this idealization process-the transition from a complex physical model 

to a simpler one but incorporating all the relevant physics-is a mathematical model consisting of 

a number of equations that closely represent the behavior of the physical model. A formal seven-step 

process was proposed by Morris and Vignjevic [ 151. An experienced analyst will go through the 

procedure guided by a few principles and much insight gamered over a long career. The time 

required might take weeks to months, depending on the complexity of the physical system and the 

accuracy required in the analysis. If the assumptions of the mathematical model are reasonable, very 

7 
little mathematical modeling error results. If this is poorly done, say if the height-to-span ratio of 

beam theory is violated, the thickness-to-radius ratio for thin shells is not satisfied, a poor choice of 

material properties is made, or the constitutive model omits a critical item such as dynamic failure, 

serious errors can be incurred. Such synthesis is not taught in schools but learned in apprenticeship 

with an experienced modeler. Fortunately, in the vast majority of cases, this is done very well so that 

mathematical modeling errors are negligibly small, or at least smaller than errors committed by code 

users, the topic of the next section. 

A solution is needed for the mathematical model. Since most problems involving 

high-strain-rate loading are not analytically tractable, recourse is made to a computer. In the process 

of computing, especially in matrix operations, situations occur where differences between numbers 

of almost equal magnitude must be taken. This can lead to situations where the roundoff error can 

completely overwhelm the computed quantity. Given a machine, there is always a limit to the mesh 

refinement beyond which computed quantities may be 100% erroneous [ 111. Roundoff errors may 

be kept negligibly small by using “ . ..longer wordlength machines and double precision arithmetic 

with not too refined finite element meshes” [ 111. 

Most dynamic analyses for problems involving wave propagation are done with the simplest 

elementwonstant or linear strain triangles and quadrilaterals in 2-D computations, tetrahedra and 

hexahedra in 3-D computations. Many early (mid-70s) finite-element codes, such as Lawrence 

Livermore National Laboratory’s DYNA and DYSMAS-L, incorporated high-order elements in their 

initial formulations. With experience, these were dropped. Many transient response calculations 

in the wave propagation regime involve the presence of steep stress gradients and shock waves. It 

has been found with experience that there is marginal increase in accuracy but considerable increase 

in cost in trying to model what are essentially discontinuities with higher order polynomials. The 

characteristics of these elements, and the problems of locking and reduced integration associated 

with them, are lucidly discussed in the book by MacNeal [ 141. The original literature and the code 

manuals should also be reviewed in order that these elements not be misused, 

8 
Code users should also be aware of the viscosity built into implicit temporal integrators and the 

conditional stability of explicit integrators. Additional information can be found in Belytschko and 

Hughes [ 171, Donea [ 181, and Zienkiewicz [ 193. 

4. The Human Factor 

It is generally accepted that, if the idealization from physical system to mathematical model is 

done well, the errors due to truncation, roundoff, and other properties of finite-element or 

finite-difference schemes can be readily detected and contribute to no more than about 5% of the 

total solution error. Solutions, however, can be totally invalidated by poor choice of mesh, failure 

to include relevant physics in the constitutive description, using a limited or inappropriate database 

to evaluate the constants of a constitutive model, failure to recognize instabilities, or the effects of 

contact surfaces on numerical solutions. In short, computational techniques come with certain 

built-in limitations that are easily recognizable and, with few exceptions, controllable. To really 

mess up requires a human. 

Some, but hardly all, of the errors in the application of computer codes to practical problems 

involve the following. 

Meshing: A code user has available a wide choice of elements in any commercial code. 

Having selected one or more, the user can then vary element aspect ratio, the arrangement 

of elements or element groups, choose uniform or variable meshing, and even introduce 

abrupt mesh changes. All of these will influence the solution to some degree. 

Constitutive Model: Assuming an appropriate model has been selected to account for 

material behavior under high-rate loading, criteria for material failure, and descriptions of 

post-failure behavior, the problem of selecting values for the various material constants in 

the constitutive model remains. These must be selected from experiments conducted at strain 

rates appropriate to the problem. In many cases, data may not be available. This is 

9 
particularly true for situations involving material failure. The estimation of these factors then 

depends on the skill, knowledge, and experience of the user, and computed results will vary 

accordingly. 

Contact Surfaces and Material Transport: Lagrangian codes incorporate a wide variety of 

algorithms to account for contact-impact situations. Eulerian codes have a variety of 

methods for determining the transport of material from one cell to another. Each algorithm 

uniquely affects the solution. Some codes have incorporated a variety of algorithms and 

allow the user a choice. The burden is then on the user to choose wisely, and this cannot be 

done without a knowledge of how the various algorithms affect the solution of both global 

(e.g., displacements) and local (e.g., strain) variables. 

Shortcuts: Because of the expense involved in 3-D calculations, recourse is sometimes made 

to plane-strain solutions. These are 2-D approximations of 3-D phenomena. Sometimes they 

produce excellent results, sometimes disasters. 

This report focuses on problems involving meshing. Subsequent reports will address the other 

topics. 

5. Problems Related to Computational Meshes 

Theoretically, the ideal mesh is uniform in all coordinate directions and “converged” for the 

critical variable for the problem; that is, it is small enough to give accurate results so that further 

refinement dramatically runs up the cost of computing with negligible improvement in accuracy. 

In practice, especially when performing 3-D calculations, this goal is impossible to achieve. 

Compromises must be made, and the educated analyst must know what effect these compromises 

have on the numerical solution to his/her problem. In short he/she must know the physical problem 

and understand how compromises to the computational mesh affect the solution. 

10 
There are a number of factors that affect mesh integrity. These include element aspect ratio, 

element arrangement, uniform vs. graded meshes, and abrupt changes in meshes. Each is now 

looked at in turn. 

5.1 Element Aspect Ratio. Ideally, calculations would be done with 1: 1 aspect ratios in the 

elements. This hardly ever happens in practical 2-D and 3-D calculations. Thus, it would be nice 

to know how the solution is affected when elements with aspect ratios exceeding 1: 1 are used. 

Creighton [20], using the EPIC-2 code, looked at the effects of aspect ratio, artificial viscosity, and 

triangular element arrangement on elastic and elasto-plastic impact situations. EPIC uses 

constant-strain triangular elements that can be arranged in a number of ways. The possibilities 

included in Creighton’s study are shown in Figure 2. The elastic calculations were performed for 

a steel bar with length-to-diameter (IJD) ratio of 100 striking a rigid barrier at 3.048 m/s. Numerical 

solutions were compared with an exact solution by Skalak [21], Figure 3, which takes into account 

the effects of radial inertia. Calculations were performed with one (400 elements), two 

(1,600 elements), and three (3,600 elements) crossed triangles (four triangles per quadrilateral) 

across the radius of the bar (length, L - 12.7 cm; diameter, D - 0.254 cm). All calculations were 

done with an aspect ratio of 1: 1. The axial force in the rod as a function of position was compared 

with Skalak’s solution at various times. 

As expected, the results show the grid acting as a frequency filter. As the grid is resolved, more 

and more ringing is computed behind the wavefront. Enough high-frequency components are 

accounted for in the 3,600-element calculation to compare very favorably with the analytical 

solution. 

Holding the number of elements across the bar radius at three, the element aspect ratio was now 

increased from 1: 1 to 4: 1. Again, the filtering characteristics of the mesh is clearly shown. 

High-frequency components are gradually suppressed until, with the 4: 1 mesh, only the ringing 

directly behind the wave front remains (Figure 4). 

11 
(a] 1DIA = 1 

(b) JDIA=2 

Hypotenuse of the triangle is drzwn from the 

lowsr M hand corner to the upper right lmnd 

M mer, 

Hypoterruscefthetriangleisd~awnfromthe 

uppr left hand mrnm to the Lower right tend 

cw rner_ 

(c) Quadrikterlal elements Fo mpriti of fo ur t ria @es, 

Figure 2. Element Orientations in the EPIC-2 Code. 

12 
. 

---- ELEMENTARY SOLUTION 

- SOLUTION WITH RADIAL 

(SKALAK 1957) 

INERTIA 

DISTANCE FROM END OF ROD 

Figure3. Comparison of Elementary and Skalak Solutions for Bar Impact (Skaiak Solution 

Is Drawn Freehand). 

13 
Figure 4. Signal Filtration With Mesh Aspect Ratio. 

14 
5.2 Element Arrangement. The arrangement or mixing of elements within a computational 

mesh can be a problem. Figure 2 shows three possible arrangements of triangular elements. This 

aspect of mesh generation was also investigated [20]. The specific problem considered was the 

impact of a steel sphere with a 5.08-cm-thick aluminum target at a velocity of 1,524 m/s. The 

two-triangle orientations (Figure 2[a] and [b]) gave rise to asymmetries in the calculation. 

Depending on the orientation of the diagonal, results were either too stiff or too soft. Optimal 

performance was achieved using the four-triangles-per-quadrilateral grouping (Figure Z[c]). Similar 

results had been obtained by Zukas [22]. The definitive study on element arrangement for accurate 

elasto-plastic solutions was done by Nagtegaal et al. [23]. 

Johnson and coworkers [24,25] observed that, for 2-D triangular elements, accuracy is much 

improved if the elements are arranged in a crossed-triangle (four triangles within a quadrilateral) 

arrangement or if a single average pressure is used for each group of two adjacent triangles (two 

triangles within a quadrilateral). In three dimensions, Johnson et al. [25] performed a number of 

simulations using configurations of 6 tetrahedral elements within a brick, 6 tetrahedral elements per 

brick using a single average pressure for all 6 tetrahedra, as well as an arrangement of 24 tetrahedral 

elements per brick. This latter case, though the most costly configuration, was expected to produce 

the best results since the asymmetries present in the six tetrahedra arrangement would be absent. It 

was also expected that the six tetrahedra configuration with pressure averaging would perform better 

compared to the six tetrahedra configuration using individual element pressures due to the reduction 

in the number of incompressibility constraints [23]. Some test cases were compared to experimental 

data [26]. Between 37,000 and 39,000 elements were used for these calculations of an UD = 10 

tungsten rod striking a 2.5”cm steel plate at normal incidence with a velocity of 1,520 m/s. The case 

of the six tetrahedra without pressure averaging produced the closest correlation with experiment for 

residual velocity and the worst for residual rod length. Furthermore, there was indication of locking 

in the target grid. The closest correlation for rod residual velocity and residual length was achieved 

for the symmetric 24 tetrahedra per brick arrangement. However, in all cases, including a 2-D 

calculation, EPIC tended to overpredict residual length by about 16%. This could be due to a 

number of factors not necessarily related to the grid. 

15 
A number of papers present comparisons of plastic strain profiles for Taylor cylinder 

impacts-the impact of a deformable, short cylinder (UD < 3, generally, although longer rods have 

been used) striking a rigid surface at impact velocities under 100 m/s. Johnson [24], for example, 

compares DYNA, NlKE, and EPIC-2 results with various arrangements of triangles (Figure 5). 

Many more papers have since appeared to not only test element formulations and arrangements but 

constitutive relations as well, Such results are interesting but tend to show only minor differences 

for the various cases considered. It is now clear that the Taylor cylinder is not a very sensitive 

measure of element arrangement effectiveness. Neither is it a good discriminant for testing 

constitutive models. 

5.3 Uniform and Variable Meshes. The effects of mesh size on wave propagation problems 

can be seen in Figures 6 and 7. These depict the impact of a long S-7 tool steel rod (IfD = 10, 

hemispherical nose, D = 1.02 cm) into 2.56-cm-thick RHA plate at 1,103 m/s. The characteristic 

lengths of this problem are the target thickness and projectile radius. For good results, there should 

be at least three elements across the radius of the rod. Since uniform mesh spacing is the ideal, this 

governs the number of elements to be used in the projectile and target. 

Figure 6 shows initial grids and results at 50 ms after impact. One, three, and five 

crossed-triangle meshes were used across the projectile radius for the coarse, medium, and fine cases, 

respectively, The target grid was then set by requiring a 1: 1 aspect ratio of all elements in the 

calculation. The coarse grid computation,shows some anomalies near the projectile-target interface 

and a v-shaped crater, indicative of numerical difficulties with triangular elements. The other two 

calculations (Figures 6[d] and [e]) appear to be reasonable. The experimentally determined 

projectile residual mass was 32.1 g, and the residual velocity was 690 m/s. The computed residual 

masses were 29 g, 36 g, and 37 g for the coarse, medium, and fine grids, respectively, while the 

residual velocities were 600 mls, 670 m/s, and 680 m/s. Coarse zoning is adequate for “quick and 

dirty” scoping calculations or to get preliminary estimates of global quantities such as length loss 

in the rod, approximate hole size, residual velocity, and overall deformed shapes of the two solids. 

It is, however, inadequate to resolve strain and pressure fields with any degree of accuracy. If the 

16 
CCIPPER ROD 

c * 3.24 ;m 

P -432 ;m 

v .z?l wi 

CM c 

DYWU lHAlloUl ST1 

CASE 0 

EPIC -2 

i 

am 

- ROD HO1 AllOMD 10 SEPARAR FROM SURFACC’ 

-- PLASTIC SIRAM tOMOuRS SHOW At IMltRvALS OF ab 

CAY B 

EPIC .? 

w 

EPIC - 2 

i 

am 

Figure 5. Effects of Element Arrangement on Plastic Strain Distribution [24]. 

17 
-6. 

t 

-8. 

-18. , 

-12. -la. -8. 4. -4. -2. 8. 2. ,. 6. m. IQ. a, 

(a) Coarse Grid - Initial 

4. 

t 

-18. , , . , , , 

-12. -18. -u. -6. 4. -2. 8. t. 4. 6. 8. iti. It. 

(c) Medium Grid - Initial 

(e) Fine Grid - Initial 

4.. 

2.. 

8.- 

-2: 

-4: 

(b) Coarse Grid - Deformation at 50 p, 

6.. 

2.. 

4: 

-8. 

-1. , , , , , , , , -It. -18. 4. 4. 4. -z. 8. z. 4. 6. 0. 18. 12. 

(d) Medium Grid - Deformation at 50 ps 

-6. 

4: , , , 

-12.8 -18.8 4.0 -6.3 4.n -2.8 8.8 2.B 4.0 6.0 B.B r.0 u.* 

(f) Fine Grid - Deformation at 50 ps 

Figure 6. Grid Effects on Long-Rod Penetration. 

18 
10 

6 

graded mesh cycle= es7 

4. -2. 8. 2. 4. 6. 8. 

(a) Gradually Changing Mesh. 

I Tim== S.eaE-85 abrupt grid chanpc cycle= 713 

a. 

6.' 

4. 

2. 

8. 

-2. 

I 

-18. 4. -6. 4. -2. a. 2. 4. 6. 8. 

10. 4 

(b) Abrupt Grid Changes. 

Figure 7. Gradual vs. Abrupt Grid Changes. 

19 
calculation employs failure criteria based on stress wave profiles, results from such coarse 

calculations will be meaningless. 

As spatial resolution is increased by a factor of 2 for 2-D simulations, computer time rises by 

a factor of 8 for explicit methods [27]. Thus, the resolution used should match the accuracy 

required. If comparison is to be made with time-resolved pressure, stress, or strain data, if internal 

failure (such as spall) occurs, fine resolution is required for a meaningful calculation, even though 

the cost is high. Compromising the accuracy of the calculation to save money in these cases is not 

justifiable since maximum savings come from not doing the calculation at all. This has the added 

advantage of not generating meaningless numbers that someone not familiar with the events of the 

calculation might be tempted to believe. On the other hand, if only global (integral) data are 

available for comparison (residual masses, lengths, hole sizes, and deformed shapes), a reasonably 

crude and inexpensive calculation can be done, provided its interpretation is not pushed too far. 

It is possible to take advantage of the localized nature of impact problems when setting up 

computational grids. Typically, the high strain rates and the high pressures, strains, and temperatures 

that accompany them are corrEmed to a narrow region, or process zone, that extends about three to 

six striker diameters from the impact interface, depending on striking velocity. Figure 7(a) shows 

results for a calculation that takes advantage of this information by localizing fine zones within three 

diameters of the impact interface and then gradually expands element size in regions where, at most, 

elastic waves will propagate. The result is accuracy comparable to the uniformly gridded case but 

with considerably fewer elements and, therefore, considerable savings in computer time. Care must 

be taken not to change element sizes too rapidly, however. A change in element size represents a 

change in element stiffness even if material properties remain the same. Traveling waves, when 

encountering this stiffness difference, will act as if an impedance mismatch had occurred. Part of 

the wave will be reflected, part transmitted. If element-to-element size variation is kept below 1 OS, 

acceptable results can generally be achieved. Figure 7(h) shows the negative aspects of drastic 

changes in element size. 

20 
Using a minimum of three continuum elements across a critical dimension such as a rod radius 

or plate thickness turns out to be a good rule of thumb for Lagrangian calculations. Eulerian 

calculations require considerably more [28 J. Some problems, however, such as hypervelocity impact 

or self-forging fragment formation, may require much more. .These are situations where severe 

pressure gradients exist and move with time. Also situations such as spall, where material failure 

occurs due to the interaction of stress waves with geometric boundaries, material interfaces, or each 

other, will require fine resolution. Zukas et al. [29], studying explosively formed penetrator (EFP) 

formation and penetration, found that five to six elements through the liner thickness and fine zoning 

in the target were required to match code calculations with experiments. Melosh et al. [ 301 modeled 

dynamic fragmentation on a laboratory scale. They developed a fragmentation model and found 

good correlation with a wealth of experimental data for the largest fragment size and the fragment 

size-number distribution, provided that an adequate numerical resolution was used. Resolutions of 

12 x 24 cells (where 12 cells defined target radii, which ranged from 2-12 cm) were-used for most 

calculations. Resolutions of 6 x 12 and 24 x 48 cells were also used. All three resolutions gave 

about the same result. However, substantially finer grids (40 x 80) were needed to match observed 

near-surface spallation. Johnson et al. [25] also looked at the effects of grid size on fragment 

distribution for normal impacts of copper rods at 2 km/s against 1 -cm steel plates. Calculations were 

performed using 1,600 (Case A) and 4,096 elements (Case B). Figure 8 shows the effects of 

gridding on fragment size distributions for the different grid sizes. 

Johnson and Schonhardt [3 l] investigated the sensitivity of the EPIC-2 and EPIC-3 codes to a 

number of factors, including gridding for normal and oblique incidence problems. Normal impact 

calculations for an UD = 10 tungsten rod striking a 2.5-cm steel plate at a velocity of 1.5 km/s were 

made with 144,576; 1,294; and 2,304 elements. Oblique impact calculations at 60”, measured from 

the target normal, were made with 480; 1,920; 4,320; and 7,680 elements. With the exception of the 

lowest resolution, there was relatively little difference for the higher resolution calculations in terms 

of residual velocity, residual mass, and hole diameter for the 2-D calculations. Significant increases 

in CPU times were observed as resolution was enhanced. Similar results were obtained for the 3-D 

simulations (Figure 9) comparing residual masses, velocities, hole diameters, rotational velocity, and 

deflection). 

21 
CASE A (BASELINE) CASE B (FINER GRID) 

1, 

,600 ELEMENTS 

ICM Am 4000 ELEMENTS 

-CASE h IBASELINE) 

FRAGMENT SIZE ,mm) 

Figure 8. Fragment Size Distribution vs. Grid Size [25]. 

22 
RESULTS AT 40 p.s 

COARSE GRID 

RESULTS AT 120 ps 

u* = 0.66 M’ = 0.53 

D’ = 32 D’ * 3.6 

to = o.ors%m a, = o.iawps 

Figure 9. Grid and Contact Surface Effects for Oblique Penetration [31]. 

23 
Zoning requirements for 2-D and 3-D Eulerian calculations with the CTH code for long-rod 

penetration problems are considered in Littlefield and Anderson [28]. 

Guidelines for determining resolution and material modeling for practical engineering problems 

are given in the report of the National Materials Advisory Board Committee titled “Materials 

Response to Ultra-High Loading Rates” [ 161. The committee recommended an iterative procedure 

of successive refinements involving computations with existing relatively simple failure descriptions, 

dynamic material characterization employing relatively simple and standardized techniques, and 

experimentation to produce useful results for design purposes in many applications. Their report 

suggests that 

. . . rough computations, using simple material models with published or even 

estimated material properties, may be used in conjunction with exploratory test 

firings to scope an initial design. Comparison of test data with predictions may 

reveal discrepancies which suggest refinements in the computations or material 

models, and the need for some material property measurements. Once reasonable 

agreement has been achieved, another round of computations may then be performed 

to refine the design. Test firings of this design might use more detailed diagnostic 

instrumentation. This sequence is iterated, including successively more detail in 

computational models, material property tests, and ordnance test firings, until a 

satisfactory design is achieved. In this procedure, unnecessarily detailed 

computations, material property studies or test firings are minimized; only those 

details necessary to achieve a satisfactory design are included. 

5.4 Abrupt Changes in Meshes. Abrupt grid changes are common in statics calculations. The 

stress gradient is fixed in space. A grid sufficiently fine for all practical purposes is superimposed 

over the region. The remainder of the physical body is then modeled with rather large elements to 

account for the total mass and boundary conditions. 

24 
In dynamics calculations, the location of the stress or pressure gradient is a function of time as 

well as space. It cannot be overemphasized that wave propagation governs the response, Recall 

from wave propagation theory that stress waves are reflected from material interfaces, geometric 

boundaries and each other. Partial reflection also occurs at internal points of discontinuity. 

Chapter 10, in Fried [32], describes this very lucidly, as do Bazant and his colleagues [33-351 for 

a number of practical systems. A spurious reflection 

. . . occurs when the traveling wave crosses over from a fine mesh region to a coarse 

mesh region. A given mesh size has a lower limit to the wavelength it can 

approximate or an upper limit to the frequency it can transmit - the cutoff frequency. 

When the wave enters the coarse mesh region, some of its high frequency 

components cannot penetrate and are reflected . , . . As the wave reaches the larger 

elements, small waves appear, traveling backwards. Also, because of the largest 

element size, dispersion becomes more pronounced in the form of leading small 

waves in front of the original one [32]. 

In Fried [32], these points are illustrated with examples of waves in strings. Consider also the 

following problem, where an UD - 3 mild-steel projectile impacts an armor-steel target at a velocity 

of 0.8 km/s with sharp mesh discontinuities in both projectile and target, Figure 10 shows the grid 

and results at intermediate times. Figure 11 shows the end result. Note that the grid discontinuity 

has, in effect, predetermined the outcome. The elements are sufficiently large that the distortions 

in the fmely gridded zone are enhanced by reflected high-frequency waves from the fme/coarse 

element boundary. Compare the results of Figure 12(a) with those of 12(b), where a graduated mesh 

was used, and Figure 12(c), with a uniform mesh. Keep in mind that, in high-velocity impact 

problems, the most severe distortions occur within three to six projectile diameters, depending on 

impact velocity. To the extent possible, this region should be uniformly zoned, with element size 

gradually increased by no more than 10% (folklore) from there onward. 

25 
-4: 

4. 4. 4. -3. -2. -a* P. 1. 2. 3. 4. 5. 6. 

(a) 0 ms 

(b) 5 P 

,.. 

3. 

1.’ 

1: 

1.. 

-1.. 

4.. 

-3: 

-i.- 

-9.c 

-l.- 

-6. -5. 4. -3. -2. -L. 8. 1. 2. 3. 1. 6. 6. 

(4 10 p 

4.. 

3.” 

2.’ 

1.’ 

1.. 

-I_. 

-2. 

-3: 

+.- 

Figure 10. Abrupt Grid Change Effects on Penetration. 

26 
The= 2.58E-8s 

4 .- 

Test 5: hbrupt Grid Changes cL$le= 727 

3 

0 

2 

‘. 

. 

. 

:. 

1 1.’ 

9 ‘. - 

-1 *a- 

-2 :. 

1 1 1 1 L 1 1 1 1 I 1 I 

-6. -5. 4. -3. 4. -1. 8. 1. 2. 3. 4. 5. 6. 

, 

Figure 11. Residual Penetrator and Hole Size in the Presence of an Abrupt Grid Change. 

27 
4.. 

1.. 

z: 

I: 

..- 

-1.. 

-2. 

-3: 

4: 

4. -5. 4. -3. -2. 4. I, 1. z. 3, ,. 5. 6. 

(4 

(W 

Figure 12. (a) Abrupt Grid Change, (b) Gradual Change in Element Size, and (c) Uniform 

Meshing. 

28 
6. Summary 

The causes of disagreement between large-scale code calculations and reality for problems 

involving fast, transient loading have been discussed. The experience of the analyst is a prime factor 

in the successful use of commercial codes for problems in dynamics. The analyst needs an 

appropriate educational background and a keen understanding of the physics and mechanics of the 

problem being addressed. He/she must also have sufficient experience with numerical techniques 

and large-scale computations in order to select the appropriate computational tool for the problem 

and to evaluate the computational results. The currently available commercial codes for dynamics 

problems are in no way “black boxes” that can be assigned to junior engineers with demands for 

immediate production. The consequences of inappropriate analyses can range from embarrassment 

and loss of funding through catastrophic failure of poorly designed structures under service loads, 

with the liabilities and litigations that inevitably follow. 

Y 

29 
30 
7. References 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

13. 

14. 

Zukas, J. A., and W, P. Walters (editors). Explosive Effects and Applications. New York: 

Springer-Verlag, 1998. 

Cooper, P. W. Explosives Engineering. New York: VCH-Wiley Publishers, 1996. 

Cooper, P. W, and S. R. Kurowski. Introduction to the Technology of Explosives. New York: 

VCH-Wiley Publishers, 1996. 

Cheret, R. Detonation of Condensed Explosives. New York: Springer-Verlag, 1993. 

Blaqnski, T. Z. (editor). Explosive Welding, Forming and Compaction. London: Applied 

Science Publishers, 1987. 

Fickett, W., and W. C. Davis. Detonation. Berkeley: University of California Press, 1979. 

Mader, C. L. Numerical Modeling of Detonation. Berkeley: University of California Press, 

1979. 

Mader, C. L. Numerical Modeling of Explosives and Propellants. Boca Raton: CRC Press, 

1997. 

Herrman, W. “Current Problems in the Finite Difference Solution of Stress Waves.” Nonlinear 

Waves in Solids, Proceedings of the Workshop at University of Illinois, Chicago, IL, 

21-23 March 1977. 

Szabo, B. A., and R. L. Actis. “Finite Element Analysis in Professional Practice.” Computer 

Methods in Applied Mechanics and Engineering, vol. 133, pp. 209-228, 1996. 

Utku, S., and R. J. Melosh. “Solution Errors in Finite Element Analysis.” State-of-the-Art 

Surveys on Finite Element Technology, A. K. Noor and W. D. Pilkey (editors), New York: 

ASME, 1983. 

Melosh, R. J., and S. U&u. “Principles for Design of Finite Element Meshes.” State-of-the-Art 

Surveys on Finite Element Technology, A. K. Noor and W. D. Pilkey (editors), New York: 

ASME, 1983. 

Meyer, C. (editor). Finite Element Idealization. New York: ASCE, 1989. 

MacNeal, R. H. Finite Elements: Their Design and Performance. New York: Marcel Dekker, 

1994. 

31 
15. 

16. 

17. 

18. 

19. 

20. 

21. 

22. 

23. 

24. 

25. 

26. 

27. 

Morris, A. J., and R. Vignjevic. “Consistent Finite Element Structural Analysis and Error 

Control.” Computer Methods in Applied Mechanics and Engineering, vol. 140, pp, 87-108, 

1997. 

National Materials Advisory Board. “Materials Response to Ultra-High Loading Rates.” 

NMAB-356, Washington, DC, 1980. 

Belytschko, T., and R. J. R. Hughes. Computational Methods for Transient Analysis. New 

York: Elsevier Science Publishing Co, Third Printing, 1992. 

Donea, J. (editor). Advanced StructuralDynamics. London: Applied SciencePublishers, 1978. 

Zienkiewicz, 0, C. “Finite Elements in the Time Domain.” State-of-the-Art Surveys on Finite 

Element Technology, A. K. Noor and W. D. Pilkey (editors), New York: ASME, 1983. 

Creighton, B. “Numerical Resolution Calculation for Elastic-Plastic Impact Problems.” 

BRL-MR-3418, U.S. Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD, 

1984. 

Skalak, R. “Longitudinal Impact of a Semi-Infinite Circular Elastic Bar.” Journal of Applied 

Mechanics, Transactions ASME, vol. 24, pp. 59-63, 1957. 

Zukas, J. A. “Some Problems With the EPIC-2 Code.” Internal Memorandum IMR 647, 

U.S. Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD, 1979. 

Nagtegaal, J. C., D. M. Parks, and J. R. Rice. “On Numerically Accurate Finite Element 

Solutions in the Fully Plastic Range.” Computer Methods in Applied Mechanics and 

Engineering, vol. 4, pp. 153-178, 1974. 

Johnson G. R. “Recent Developments and Analyses Associated With the EPIC-2 and EPIC-3 

Codes.” Advances in Aerospace Structures and Materials, S. S. Wang and W. J. Renton 

(editors), vol. AD-01, New York: ASME, 1981. 

Johnson, G. R., R. A. Stryk, T. J. Holmquist, and 0 A. Souka. “Recent EPIC Code 

Developments for High Velocity Impact: 3D Element Arrangements and 2D Fragment 

Distributions.” International Journal of Impact Engineering, vol. 10, pp. 281-294, 1990. 

Hohler, V., E. Schneider, A. J. Stilp, and R. Tham. “Length and Velocity Reduction of High 

Density Rods Perforating Mild Steel and Armor Steel Plates.” Proceedings of the 4th 

International Symposium on Ballistics, Monterey, CA, 1978. 

Zukas, J. A., and K. D. Kimsey. “Supercomputing and Computational Penetration Mechanics.” 

Computational Aspects of Contact, Impact and Penetration, R. F. Kulak and L. E. Schwer 

(editors), Lausanne, Switzerland: Elmepress International, 1991. 

32 
28. Littlefield, D. L., and C. E. Anderson, Jr. “A Study of Zoning Requirements for 2-D and 3-D 

Long-Rod Penetration.” American Institute of Physics, pp. 1131-l 134, 1996. 

29. Zukas, J. A., C. A. Weickert, and P. J. Gallagher. “Numerical Simulation of Penetration by 

Explosively-Formed Projectiles.” Propellants, Explosives and Pyrotechnics, vol. 18, 

pp. 259-263,1993. 

30. Melosh, H. J., E. V. Ryan, and E. Asphaug. “Dynamic Fragmentation in Impacts: Hydrocode 

Simulation in Laboratory Impacts.” Journal of Geophysical Research, vol. 97, no. E9, 

pp* 14735-14759,1992. 

3 1. Johnson, G. R., and J. A. Schonhardt. “Some Parametric Sensitivity Analyses for High Velocity 

Impact Computations.” Nuclear Engineering and Design, vol. 138, pp. 75-9 1, 1992. 

32. Fried, I. Numerical Solution of Dijferential Equations. New York: Academic Press, 1979. 

33. Bazant, Z. P. “Spurious Reflection of Elastic Waves in Non-Uniform Finite Element Grids.” 

Computer Methods in Applied Mechanics and Engineering, vol. 16, pp. 91-100, 1978. 

34. Bazant, Z. P., and Z. Celep. “Spurious Reflection of Elastic Waves in Nonuniform Meshes of 

Constant and Linear Strain Finite Elements.” Computer and Structures, vol. 15, pp. 45 l-459, 

1982. 

35. Celep, Z., and Z. P. Bazant. “Spurious Reflection of Elastic Waves Due to Gradually Changing 

Finite Element Size.” International Journal for Numerical Methods in Engineering, vol. 19, 

no. 5, pp. 631446,1983. 
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42 
>ractical Aspects of Numerical Simulations of Dynamic Events: 

tieshing 

. Zukas’ and Daniel R. Scheffler 

J.S. Army Research Laboratory 

4Tl-N: AMSRL-WM-TC 

4berdeen Proving Ground, MD 21005-5066 

REPORT NUMBER 

ARL-TR-2303 

II. SUPPLEMENTARY NOTES 

*Computational Mechanics Consultants, Inc., P.O. Box 11314, Baltimore, MD 21239-0314 

128. DlSTRlBUflONlAVAllABlUTY STATEMENT 

Approved for public release; distribution is unlimited. 

12b. DISTRIBUTION CODE 

13. ABSTRACT(Mexlmum pw words) 

The use of finite-difference and finite-element computer codes to solve problems involving fast, transient loading I 

commonplace. A large number of commercial codes exist and are applied to problems ranging from fairly low t 

extremely high damage levels (e.g., design of containment structures to mitigate effects of industrial accident; 

protection of buildings and people from blast and, impact loading; foreign-object impact damage; and design of spat 

structures to withstand impacts of small particles moving at hypervelocity, a case where the pressures generated excee 

material strength by an order of magnitude). But, what happens if code predictions do not correspond with reality? Th 

report discusses various factors related to the computational mesh that can lead to disagreement between computatior 

and experience. Subsequent reports will focus on problems associated with contact surfaces and material transpo 

algorithms, constitutive models, and the use of material data at strain rates inappropriate to the problem. It is limited I 

problems involving fast, transient loading, which can be addressed by commercial finite-difference and finite-elemel 

computer codes. 

This report has been accepted for publication in a future volume of the International Journal of Zmpact Engineerin) 

14. SUBJECT TERMS 15. NUMBER OF PAGES 

hydrocodes, finite elements, finite difference, high velocity impact, shock waves, meshing 47 

16. PRICE CODE 

17. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 19. SECURlTY CLASSIFKATION 20. LIMITATION OF ABSTRAC1 

OF REPORT OF THIS PAGE OF ABSTRACf 

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44 
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