Structural Analysis of Underwater Detonations

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Structural Analysis of Underwater Detonations

Edvin Sjöstrand

Mechanical Engineering, master's level 2021

Luleå University of Technology

Department of Engineering Sciences and Mathematics

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Preface

I have during the spring of 2021 completed my master thesis in collaboration with SAAB Dynamics AB. It has been both a very interesting and educational period of time and therefore I want to start by expressing my gratitude to my mentor Mikael Holmgren and section manager Hans Bergman at SAAB for all the support and giving me the opportunity to investigate this project. I want to thank Pär Jonsén, my examiner at Luleå University of Technology, for all the help and support during my thesis. I also want to thank David Fyhrman at Dynamore Nordic AB for supporting me with his knowledge of the LS-Dyna software.

At last I want to give my appreciation to all my friends and family who have supported me during this spring but also during my ﬁve years at Luleå University of Technology.

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Abstract

The knowledge how an object withstand an underwater detonation is critical within the defense industry. This is mostly done today with physicals test which are both time consuming and connected with high costs. The aim of this thesis is to provide recommendations and guidelines on how to model and analyze a structural response of underwater detonations. This investigation are focused on ﬁrstly investigate several theoretical simulation methods and thereafter develop a model of the chosen method.

The simulation method was decided to be the Multi-Material Arbitrary Lagrangian Euler(MMALE) using the software LS-Dyna. To receive a model with functionality to simulate an explosion a method of six steps is developed to increase the complexity. The ﬁnal step is to be able to analyze a structural response of an object.

The validation phase contained several convergence studies of the two Equations of states and a varying element size compared to analytical equations. The plan was to perform a validation test but because of travel restrictions due to the Covid-19 situation an alternative validation method was used. This method involved two external reports with speciﬁed measurement data.

The aim to develop a model is reached as the model performs well against the cylinder in the validation phase, however the element size is the most important parameter in an accurate model. The developed model shows good agreement regarding the structural response of an object when compared to well deﬁned and reported experiments.

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Abbreviations

FEM Finite Element Method EOS Equation Of State

ALE Arbitrary Lagrangian Euler

MMALE Multi Material Arbitrary Lagrangian Euler S-ALE Structural Arbitrary Lagrangian Euler SPH Smoothed Particle Hydrodynamics DEM Discrete Element Method

FSI Fluid Structure Interaction UNDEX Underwater Explosion TNT Trinitrotoluene

FOI Swedish Defense Research Agency 2D Two Dimensions

3D Three Dimensions

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1 Introduction

Saab Dynamics is a business unit in the Swedish defense company SAAB AB. One of SAAB Dynamic’s business areas is underwater products for the civil and defense markets. These products shall be able to withstand the tough operating conditions in the environment where it operates, which also includes the ability to survive underwater detonations. Normally, this is tested by physical tests. To perform physical experiments are both time consuming and connected with high costs. This necessitates to simulate underwater detonations with high accuracy in a time-eﬀective way.

An underwater detonation is an explosion created below the water surface. These explosions can be harmful to both small objects and large structures both at and beneath the surface. The explosion differ to an explosion in air, mainly because of the difference in density of the surrounding medium. Water has a higher viscosity and requires more energy to be moved away by the explosion and will therefore behave differently. Numerical methods provide an effective way of attaining detailed knowledge about underwater detonations in addition to powerful visualizations of the phenomenon. For the development of accurate numerical models, physical testing can be minimized saving both development time and cost.

1.1 Aim and objective

The objective of this thesis is to investigate available methods to simulate an underwater detonation using numerical methods. The investigation of the methods should include the aspects of accuracy, level of complexity and eﬃciency. As a result of the investigation, a model of the preferred method should be created. This model should be able to vary parameters such as the explosive’s size and weight, distance from the explosion, water depth, size of the analyzed object and if the analyzed object is interfered by other objects or the detonation is close to the surface or the seabed. The main aim of the thesis is to give recommendations and guidelines of how to model and analyze the structural response from underwater detonations.

1.2 Pre study

A simulation of an underwater explosion (UNDEX) needs to consider multiple different phases such as the almost instant detonation, a quick propagating shock wave, a pulsating gas bubble, effect to objects, reflections and cavitation at the surface[5]. There are several different simulation methods to consider according to previous investigations given below.

1.2.1 Boundary Element Method

One of the earliest methods used to simulate an UNDEX was the Boundary Element Method (BEM). Wilker

son[35] used the method to describe a bubble’s shape of an asymmetric and three dimensional model and proved the ability to simulate a bubble’s motion towards the surface. This method was improved in several steps to simulate an UNDEX until 2008 [1].

1.2.2 Arbitrary Lagrangian Eulerian Method

One very commonly used method to simulate an UNDEX is the Multi Material Arbitrary Lagrangian Euler (MMALE) method. This method is based on a combination of Euler and Lagrangian descriptions. The method is shown to handle interactions between fluids and other structures in a proficient manner according to Souli [32]. Gannon concluded that MMALE method is accurate compared to empirical data of a near-field UNDEX but needs to be improved on far-field distances [13].

The MMALE method has been improved to simplify the meshing process and resulted in the Structured Ar

bitrary Lagrangian Euler (S-ALE). This method is similar to MMALE but is generally faster in solving large objects.

1.2.3 Discrete Element Method

The methods mentioned above are all based on elements. In addition to the methods based on elements, meshless methods are using elements with no connection between two speciﬁc nodes. One interesting meshless method is the Discrete element method (DEM). The DEM method was introduced by Cundall[30] for analysis of rock-mechanic problems in 1971 and is today mostly used for granulate materials[21, 34, 12]. No earlier reports of the subject were found using this method and therefore this method was not appropriate to simulate an UNDEX.

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1.2.4 Smoothed Particle Hydrodynamics Method

One other meshless method is the Smoothed Particle Hydrodynamics method (SPH). In 1977, Lucy[25] and Gingold and Monaghan [14], invented the smoothed particle hydrodynamic (SPH) method independently. It is a mesh free, point based method for modeling ﬂuid ﬂows, which has been extended to solve problems with material strength. This method uses particles with a density and radius and is preferred when simulating large deformations. Liu et al.[23] have presented accurate result of an explosion above ground using the SPH method.

This meshless method is also proved to combine ﬂuids and solids in an eﬀective way in the simulation software LS-DYNA[36]. The SPH method is also known to be used for multiphysics simulations[19, 20]. One of the downsides with the SPH method is that it is unstable handling tensile forces.

1.2.5 Analytical methods

To decrease the calculation time to execute a simulation, a keyword in LS-Dyna has been developed to use empirically found constants to theoretically load the mesh without simulating the core of the explosion. This keyword is named *LOAD_SSA and stands for Sub-Sea Analysis. This analytical model has been used by Nawa and Just[27] with good results. LS-DYNA has an additional Boundary integral code for underwater detonation called Underwater Shock Analysis (USA), which requires an additional license. This code is devel

oped to simulate an UNDEX and how it affects an object in a cost effective way without the need to define the surrounding water. The code has been proven to significantly decrease the simulation time according to Özarmut[37] compared to the MMALE method.

1.3 Limits

This thesis will mainly focus on the MMALE method. The reason is that it is one of the most commonly used methods for an underwater detonation and was recommended by the experts at the company Dynamore Nordic. In the defense industry conﬁdentiality is important and therefore this thesis will only handle the well known explosive material Trinitrotoluene (TNT). The software used was LS-DYNA R11.1 for the simulations and LS-PrePost and Hypermesh to create the ﬁnite element models. The simulation will be focused on the explicit solver because of the short time intervals. At the start of this thesis, it was an aim to plan and execute an experiment to test and validate the simulation data. Unfortunately, due to the travel restrictions regarding the Covid-19 situation it was not possible to physically perform these validation tests.

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2 Theory

This section involves the theory of an underwater explosion and the sequences caused by the detonation, an explanation of the explicit solver in LS-DYNA, followed by an introduction of the MMALE method.

2.1 Theory of underwater detonations

An underwater detonation can be divided into diﬀerent phases according to Cole [5]. At ﬁrst when the detonation ignites, a spherical shock wave starts to propagate at a very high speed outwards from the detonation center.

At the same time as the shock wave expands, an exothermic explosion releases its energy and creates a heated bubble with high pressure. The pressure forces the bubble to expand. As the bubble expands and the volume of the sphere increases, the pressure decreases. Because of the outward movement of the bubble, it moves past the pressure equilibrium brieﬂy until the bubble pressure is overruled by the surrounding water pressure. This unequal pressure drives the bubble to compress as the pressure of the surrounding water now is greater than the inside pressure of the bubble. This phenomena of expansion and compression occurs multiple times until all energy has faded into potential energy. As the progress of the bubble develops, it starts to slowly drift towards the water surface. The bubble pulsation can be seen in Figure 1.

Figure 1: An illustration of the bubble pulsation over time. The bean shape is created during the migration phase.

A shock wave moves faster than the bubble expansion phenomena, as can be seen in Figure 2. For example, the shock wave moves out of the studied area in matters of milliseconds, as the bubble expansion - contraction phenomena occur in the second time frame, thus a 1000x factor larger time.

Figure 2: Relation between bubble and shock wave spheres expansion with included velocity vectors. Larger arrows means higher velocity

2.1.1 Laws of ﬂuids

To obtain a better understanding of the sequences of an underwater detonation, an introduction of the three governing equations describing the motion of a ﬂuid is given. All equations regarding the motion of the ﬂuid

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during an underwater detonation are based on these governing equations.

The ﬁrst equation is the conservation of mass. This equation means that a certain mass in a closed system cannot increase or decrease in the time aspect, just redeﬁne the density inside the geometry

@⇢ + div(⇢V ) = 0 (1)

@t

where ⇢ is the density, t is the time and V is the velocity vector.

The second equation is the conservation of momentum, and is based of Newton’s second law. This equation is describing that the only force affecting the specified geometry is the difference in pressure in two different nodes.

⇢@V + ⇢(V · grad)V = -gradP (2)

@t where P is the pressure.

The third equation is conservation of energy, which basically means that the total energy of a closed system is constant.

dE P d⇢

⇢ = (3)

dt ⇢ dt 2.1.2 Detonation

A mine or other explosives at sea contains one or more explosive materials. Mines are often containing a chemical combination of several materials, partly to stabilize the unstable conditions of the materials and partly to achieve the desired eﬀect. One characteristic explosive material is TNT (Trinitrotoluene). With this material, an almost instantly exothermic chemical reaction is created when a charge ignites. The exothermic reaction causes a dramatic increase of both temperature and pressure. The instant reaction maximizes the initial pressure and the reaction is completed within 0.1 microseconds. The initial, and also maximum, pressure is according to Reid [31] dependent on the ratio of the weight and distance between measure point and detonation multiplied with a constant that is dependent on what kind of charge is being used. The peak pressure equation at a certain measuring point is stated as

⇣ W ^1/3⌘^↵

Pmax = A . (4)

R

where A and ↵ are constants depending of the detonation material, W is the weight of the charge and R is the distance from the detonation to the measuring point.

2.1.3 Shock wave

When an explosive material ignites and the almost instant exothermic reaction develops, a shock wave arises.

This shock wave expands with a high velocity, even higher than the speed of sound. As the shock wave develops, the speed converges exponentially towards the speed of sound in water[31]. The pressure at the measuring point after the detonation is explained by an exponential drop of the pressure as a function of time

(t t0)/✓

P (t) = Pmaxe (5)

where e is the Euler’s number, t is the time since the detonation, t0 is the time it takes for the shock wave to arrive at the measuring point and Pmax is the maximum pressure given in equation (4). The decay time ✓ is the time for the detonation to reach a fraction Pmax/e of the maximum pressure. During lots of empirical tests and validations it is concluded that the pressure curve is accurate until this fraction. After this fraction, the pressure curve predicted by the model will start to decrease at a higher rate than an actual pressure wave described in Equations (5). The decay time is calculated as

✓ = K1W ^1/3⇣ W ^1/3⌘ R (6)

where K1 and I are material dependent constants[22]. These constants have been empirically developed for TNT by Reid[31] and are presented in Table 1.

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Table 1: Constants for TNT from Reid [31].

A [-] ↵ [-] K1 [-] I [-] K2 [-] K3 [-]

52.12 1.18 0.092 -0.185 3.383 2.064

The impulse of the shock wave can be calculated as

Z t

I(t) = P (t)dt (7)

0

and results in the impulse per unit area. This is also a way of describing the damage that can be done by the explosion under water.

2.1.4 Cavitation

The shock wave creates cavitation when in vicinity of the sea surface. The shock wave reaches the surface and like a mirror reacts with a pressure wave in the opposite direction. These two waves are merged together at some areas and causes the water to transform to gas bubbles due to the low pressure and therefore cavitation arises because of waters low ability to handle tensions. The cavitation zone is an area over the detonation center, close to the surface. This cavitation zone reaches as far as the two waves can meet and creates a C-formation with the opening downwards. This phenomena can be seen in Figure 3.

Figure 3: An illustration of the cavitation in water created by the shock wave. The cavitation is shown as the white color.

The shock wave propagates in a spherical formation along all directions and will therefore reach the surface.

When the shock wave reaches the surface as a compressed wave, it reﬂects most of the wave as a negative pressure wave which is illustrated in Figure 4.

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Figure 4: A detailed illustration of the shock wave in Figure 3. The plain line shows a direct wave and the dashed line illustrates a reﬂected wave which causes the cavitation.

2.1.5 Bubble expansion

While the shock wave is expanding rapidly, a gas bubble is created and starting to expand as a result of the reaction. The bubble is created due to the increased internal pressure, which is higher than the surrounding water. This causes a demand of expansion to prevail the inequalities. As the bubble expands and the volume increases, the pressure inside the bubble will decrease. At one point the bubble pressure and the surrounding pressure will reach equilibrium, but because of the inertia created during the expansion, the bubble will continue to expand. Once the surrounding water pressure is large enough to overcome the bubbles movement, the bubble will reach its maximum volume. As it reaches the ﬁrst bubble maximum the bubble starts to decrease to reach equilibrium since the surrounding pressure is larger than the bubble pressure.

The bubble creates an inward motion and increases the pressure during the compression of the bubble. This inward motion increases until the bubble is strong enough to overcome the inward motion and will then move in an outward motion again.

The maximum radius of the ﬁrst bubble expansion R and the time until the ﬁrst minimum T can both be described theoretically[31]. These equations varies on two variables, the detonation depth D and the explosive charge W . The equations are stated as

⌘1/3

⇣ W

Rmax = K2 (8)

D + 9.8 and

W ^1/3

T = K3 5/6 (9)

(D + 9.8)

where K2 and K3 are detonation material dependent constants given in Table 1.

This bubble phenomena can be seen in Figure 5.

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Figure 5: Two graphs of the expansion (a) and pressure (b) curves. In the upper graph, it can be seen how the bubble expands and decreases in size. As a result of gravity the bubble migrates towards the surface, mostly during the decrease phase. The lower graph shows how the pressure starts at an extreme level and then decreases below the surrounding pressure and then increases again. Both graphs are describing the same phenomena. The ﬁgure is inspired by Costanzo[8].

2.1.6 Energy balance

An explosion under water contributes to the two stated phases above. A shock wave and a pulsating bubble.

According to Reid[31], the shock wave contains 53 % of the energy released from the detonation. This shock wave losses approximately 20 %-units of the energy during the early propagation. The 33%-units of energy propagation of the shock wave is potentially harmful to an object standing in the way of the wave.

The remaining 47% of detonation energy is used to expand the ﬁrst bubble pulsation. Only 17%-units is left in the second bubble pulsation. The detonation distribution is reliable if there is no impact with the surface or the sea bottom. The distribution of the bubble migration is neglected. Generally, if a detonation is made in shallow water or close to the bottom other distributions may appear.

2.2 Finite element method

The Finite element method is a method to numerically calculate a geometry by dividing the domain into a certain number of nodes and compute them using numerical approximations. This method is one of the most commonly used methods by engineers to solve static and dynamic problems, both linear and non-linear. There are two ways to compute the displacement using a direct time integration as ﬁnite element discretization[24]. Implicit time integration solves the next time step to ﬁnd the current displacement. The implicit time integration is well known to be used at large time intervals with small to no displacements[3, 18, 16]. The other time integration is the explicit solver. The explicit solver is using the current and the step before to compute the displacements of the next time step and it is mostly used for short time intervals with large deformations. The automotive industry is a serious user of the explicit solver for crash simulations, even of formula cars [15] but also within the infrastructure, with an example of a car crashing into road bumpers[4].

2.2.1 Explicit time integration

An explicit solver is calculating the equation of motion to compute the displacement of the discrete nodes created as a discretization of the geometry[9]. The equation of the nonlinear motion system, if neglecting the damping, is

M d ¨+ fint(d) = fext (10)

where M is the mass matrix, d is the nodal accelerations, d is the nodal displacements, f¨ int is the internal forces and fext is the external forces.

The solver is using a central diﬀerence method to compute the accelerations and the velocities of the current step, using the displacement in step before and the next displacement[2]. The acceleration is stated as

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dt+ t - 2d^t+ dt t

d = ¨ (11)

t²

where the displacement of diﬀerent time steps are stated. A solvable system of the equation of motion is created by inserting Equation (11) into Equation (10). With some rearrangements the equation is stated as

dt+ t = 2dt - d^{t t}+ t²M ¹(fext - f^int(d)) (12) The explicit solver is limited to a maximum time step called the critical time step. The time step is not allowed to be larger than the critical time step and is restricted by the element length and no data is transferred to more than one element per time step. The equation for the time step is states as

le

te = (13)

ce

where le is the length of the element and ce is the speed of sound of the material for the speciﬁc element. Note that the time step t may diﬀer during a simulation using an explicit solver depending of the critical time step for the current time.

2.2.2 Lagrangian and Eulerian formulation

There are two main ways to compute the positions of the nodes for the mesh, either Euler or Lagrange conﬁg

uration[26]. Figure 6 illustrates the diﬀerences between Euler and Lagrange mesh.

The Lagrangian description contains the assigned material for the same elements at all times. If forces applies and the mesh deforms, it can lead to distortion which can aﬀect the accuracy. The boundaries applied of the nodes are stable and follows the element at all times[26].

Euler’s description uses a main mesh domain that is ﬁxed in space. Inside of the domain, the material can handle large deformations and movements, and is the focus for the description, and takes no individual node in consideration. This is mostly used to describe and simulate ﬂuids[26].

Figure 6: The two descriptions of Lagrange and Euler meshing. Lagrange

As described above, both of the descriptions have some advantages against each other. Combining both of them results in a method recognized as Arbitrary Lagrangian-Eulerian description, resulting in a more accurate analysis[10, 11].

2.2.3 Multi-Material Arbitrary Lagrangian Eulerian method

Arbitrary Lagrangian-Eulerian(ALE) method is divided into two steps while solving a problem with ALE elements[17]. The ﬁrst step involves a Lagrangian time step, shown in Figure 6, where the mesh potentially

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moves or deforms. The second step of the ALE description is to "correct" the nodes, boundaries, and the material points inside of the mesh. This correction is called advection step and minimizes the distortion created in the Lagrangian procedure. Advection is very computationally costly compared to the Lagrangian time step. As mentioned, the ALE method includes an Eulerian formulation and opens up to assign more than one material for each element. The materials movements are computed during the advection phase. Therefore, with the ability to describe multiple materials, the method is called Multi-Material Arbitrary Lagrangian Eulerian (MMALE).

2.3 The software LS-Dyna

The software LS-Dyna[17] is an advanced simulation program developed to solve FEM problems using both explicit and implicit methods. The reading file that is solved by the software is built up by so-called keywords into a Key input-file or .k file. These keywords are starting with a *-mark followed by the name of the keyword.

It could include keywords describing boundary conditions and material information to the simulation time and nodal information.

2.4 Equation of states

One way to describe a material’s properties is to use equations of states(EOS). These equations are able to define the state of a material, depending on the circumstances. One common Equation of state is the Linear Polynomial. If the material might have a non linear behavior, the EOS Gruneisen is popular to use. For the detonation material, EOS JWL High Explosive is preferred. All equations are expressing the pressure p by using the internal energy per initial volume E, with the ability to project the specific materials form. These three equations of states are defined below. More detailed information about the stated EOS can be found in LS-DYNA’s manual [7].

2.4.1 Linear Polynomial

The linear equation of state uses a linear polynomial equation to express the pressure of a material as

p = C0 + C1µ + C2µ + C2 3µ ³+ (C4 + C5µ + C6µ ²)E (14) where the constants Ci are material deﬁned. The constant µ is expressed as

µ = 1

- 1 (15)

V

where V is the relative volume, which is the current volume divided by the initial volume ._VV₀

2.4.2 Gruneisen

The Gruneisen uses a cubic polynomial equation to formulate the pressure. It is separated into two equations, one for compressed materials

0 a

⇢0C²µ(1 + (1 - ₂) - µ₂ ²)

p = ( _µ2 µ³ ) + (I⁰+ ↵µ)E (16)

1 - (S¹- 1)µ - s²µ+1 - s³(µ+1)²

and one for expanded materials

p = ⇢0C²µ(

I0 + ↵µ)

E (17)

where I0, ↵, S1, S2 and S3 are material deﬁned constants and µ is computed as in Equation (15).

2.4.3 JWL

The high explosive equation of state is calculating the pressure of detonations. It was developed by Jones, Wilkens and Lee. The pressure is calculated as

! ! !E

p = A⇣

1 - ⌘

e ^R¹^V+ B⇣

1 - ⌘

e ^R²^V+ (18)

R1V R2V V

where A, B, R1, R2, !, E0 and V0 are material deﬁned constants. This EOS keyword needs to be combined with the material keyword *Mat_High_Explosive_Burn

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2.5 Representation of the detonation phenomena

The primary way to represent the detonation phenomena of a simulation for an UNDEX is to represent the material of the geometry and let it ignite. The explosive will explode and follow the theory of an UNDEX mentioned in Section 2.1. This method requires a ﬁne mesh at the center of the detonation. In LS-Dyna, the analytical formulation predicts the impact of the detonation without the need of any water surrounding the object. The name of this keyword is *Load_SSA (Sub-Sea Analysis) and computes the movement with the ability to enforce it into the mesh as a load[6]. According to LSTCs manual, it is developed to use equation (5) and includes equation (4) and (6) combined with user deﬁned constants and material properties.

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3 Methodology

The structured process to generate a model with both accurate performance and eﬃciency is stated in six steps below. Each step is adding keywords and details to the model.

• Create a model of water with correct water pressure

• Include an object that is ﬂoating, rising or sinking, using Archimedes principle

• Create a rising bubble

• Create an underwater detonation without a structure

• Create an underwater detonation with a structure

• Create an underwater detonation using the keyword *Load_SSA

The steps created knowledge and information of the speciﬁc keywords and led to the used keywords in Section 4.2.

During the development of the model and procedure stated above several convergence studies were performed to receive information of how diﬀerent parameters play a role for the simulation model. A geometry in both 2D and 3D was created after deciding a relevant mesh geometry for the studies.

The outcome of the convergence studies in combination with the developed information during the step procedure resulted in a stable setup used to create a validation model. It was planned to perform physically tests at one of SAAB’s facilities to achieve validation data but these were unfortunately canceled due to the Covid-19 situation and the company’s resulting travel restrictions. Therefore a new direction to validate the model was created.

Two investigated case-studies(external reports) with speciﬁed measurement data of the UNDEX experiments were used.

A step by step representation of the work is visualized in Figure 7.

Figure 7: An illustration of the process used to develop the model with the validation at the end.

Step 6 was not performed and is therefore marked in red.

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4 Numerical model

The following section describes the important components of the model and the chosen mesh and used materials.

It includes motivations and descriptions of what the selected LS-Dyna keyword activates for the simulations.

4.1 Computer speciﬁcations

The computer used during the convergence studies contained an Intel Xeon W-2123 processor with four cores and a 3.60 GHz base speed. The simulations were made locally on the hard drive.

During the validation phase of the model, when larger and more computationally heavy simulations were simulated, a more powerful cluster provided by SAAB AB allowing an increased amount of CPUs were used.

4.2 Keywords

To build up the models used for the simulations several important keywords are of importance and are deﬁned below.

To be able to use the MMALE method for both shell and solid elements, the element formulation 11 (MMALE

elements) is required. This formulation is used for all elements of the mesh of all simulations performed in the thesis, with the exception of the structural shell objects. In the simulation the water is desired to be inﬁnite but it is obviously not possible to have an inﬁnite amount of elements. Therefore a part of solid elements with an ambient boundary condition is placed at the boundary. This is set for the element formulation 11 with AET

= 4. This ambient keyword is initialized with the hydrostatic water pressure at the deﬁned water depth no matter the eﬀects of the surrounding.

The water pressure can be described by Archimedes principle and diﬀers linearly with depth. The equation is stated as

Ptot = ⇢gh + Psurf ace (19)

The pressure is initialized by the keyword *Initial_Hydrostatic. This keyword has the ability to deﬁne the correct pressure of the entire mesh at the ﬁrst time step and works for the entire mesh except for the boundary.

At the boundary, the ambient layer is induced with a hydrostatic pressure via *ALE_Ambient_Hydrostatic.

Both keywords create the same pressure but are needed not to receive and create any reﬂections at the boundary.

In addition to the hydrostatic pressure, a gravity acceleration is applied to all parts to simulate the relatively low but important gravitational force. The keyword *Load_Body creates the acceleration.

The *Boundary_Non_Reﬂection is considered to be used at the boundary of the geometry but after some tests it was not applicable in combination with the wanted ambient boundary.

While using ALE elements, it is possible to create ALE-materials using the keyword

*ALE_Multi-Material_Group and assign them to the speciﬁc parts. It is of high importance to assign the materials in the correct order when using three or more ALE materials. The order needs to be the predicted materials that are going to be next to each other, as the software adds one material for the calculation at the time. An example would be an explosion in water with air in a cylinder as shown in Figure 8 where it would ﬁrst calculate multi-material group 1 (MMG1), then MMG1 + MMG2 and MMG1 + MMG2 + MMG3 at last.

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Figure 8: The recommended order to assign the materials (MMG) in LS-Dyna, to simplify the simulation and decrease the simulation time. The ﬁrst in orange is the Explosive material, next to the water in blue. The green circle is illustrating the air inside a cylinder.

If a part includes more than one material the *Initial_Volume_Fraction_Geometry is used. This is used for the spherical detonation material placed in the middle of the water mesh part.

The assigned advection method for the ALE method is set in *Control_ALE as METH=2. LS-Dyna have released one improved method for high explosive materials (METH=-2) but this method is found to work better during just the small time simulation of the shock phase and is therefore abandoned.

The pressure reading is obtained using the *Database_tracer keyword. This tracer function can be included during use of the MMALE method and can either follow the fluid or be fixed at the given point. The wanted measure point of the tracer are inserted by the global coordinates. During this project, the trace point is fixed in space at all times.

The implementation of the Fluid-Structure-Interaction (FSI) is done by using the keyword

*Constrained_Lagrange_In_Solid. This FSI keyword is able to connect ﬂuids created by an ALE- formulation to a Lagrangian shell using a penalty based method. This allows the model to include an object to stand in the way of another object or to extract the impacts applied to the object by the explosion. The manual [6] has presented a step-by-step list to minimize the leakage.

The software, LS-Dyna, locates the critical time step as the minimal time step needed for the current time and multiplies it with a factor. The factor is set to 0.9 by default to prevent instabilities. During an underwater explosion it is recommended to use an even smaller critical time step factor of 0.67 for UNDEX simulations which have been the case for all the simulations.

If there is a need to calculate the model using a Multi Parallel Processor(MPP), this means dividing the model into separate parts during the solving process, the keyword

*Control_MPP_Decomposition_Distribute_ALE_Elements is preferred to separate the diﬀerent ALE parts and is recommended to use to reduce the simulation time.

In addition, if a FSI coupling is used in the simulation, using *Constrained_Lagrange_In_Solid to optimize the MPP process, the variable OPTIMPP is recommended to be activated inside the keyword *Control_ALE.

4.3 Material

The physical properties for air and ﬂuid water used in all the simulations are given in Table 2. The keyword

*Mat_Null is used to be able to combine material properties with an equation of state of the same material.

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Table 2: Material keyword *Mat_Null and the used data to describe the air and water in combination with an Equation of state.

Density [kg/m³] Pressure cutoﬀ [Pa] Dynamic viscosity [Pa·s]

Air 1.1845 0 0.0185

Water 1000 0 0.998200

The constants used in the equation of state shown in Equation (14) are given in Table 3 for the air and the water pressure.

Table 3: EOS Linear Polynomial

C0 [Pa] C1 [Pa] C2 [Pa] C3 [Pa] C4 [Pa] C5 [Pa] C6 [Pa] E0 [Pa] V0 [-]

Air 0 0 0 0 0.4 0.4 0 253313 1

Water 101325 2.25e9 0 0 0 0 0 0 1

The other equation of state, Gruneisen, shown in Equation (16), was used with the constants deﬁned in Table 4 to calculate the water pressure.

Table 4: EOS Gruneisen

C [m/s] S1 [-] S2 [-] S3 [-] I0 [-] A [-] E [Pa] V [-]

Water 1480 2.56 -1.9086 0.227 0.5 0 0 1

The material keyword used to deﬁne the explosive is the *Mat_High_Explosive_Burn. In Table 5 the data for TNT are given, which were used during the simulations.

Table 5: Material keyword *Mat_High_Explosive_Burn

Density [kg/m³] Detonation velocity [m/s] Chapman-Jouget pressure [GPa]

TNT 1630 6930 21

The EOS to calculate the explosive pressure used for the simulations of TNT are described in Table 6.

Table 6: Equation of state data for the explosive material TNT using JWL

JWL TNT

A 371 GPa B 3.23 GPa R1 4.15 R2 0.95

! 0.3

E0 7 GPa

V0 1

4.4 Mesh

The meshes and geometries were created and generated using the software Hypermesh and then imported into LS-PrePost to create the remaining keywords into the key-ﬁle.

The reason to build a mesh as a sphere was because of the spherical expansion of the shock wave and the bubble motion. This was thought to be a good way to expand the geometry with less elements than a cubic geometry.

Already from the start it indicated that the geometry depended on the inner zone where the bubble expansion occurred. This led to the mesh given below.

The mesh is one of the most important tools to create a fast but also accurate simulation. During the convergence study a quarter of a sphere was created with two diﬀerent element size zones with quad elements varied in size from 15 mm to 150 mm. An illustration of how the sphere was used can be seen in Figure 9. The inner zone

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was created big enough to include the bubble expansion, but as can be noticed in the ﬁgure, all elements will not be squared outside the geometry of the sphere. The outer zone was created to calculate the pressure with less accuracy as compared to the inner zone.

Figure 9: An illustration of the mesh for the convergence study. The length of a is 2.25 meter and b is 0.75 meter.

The 3D ﬁnite element model used for the convergence studies is visualized in Figure 10. It is built up using only solid hexa-elements.

Figure 10: The ﬁnite element mesh used for the 3D simulations. The blue color visualizes the water part and the red is the ambient boundary water part.

A similar geometry as shown in Figure 9 is used for the 2D axisymmetric simulation cases. This is created to show the eﬃciency of the 2D simulation with respect to the 3D model. The axisymmetric model needs to be placed with the symmetric line at Y-axis, where the 2D-ALE elements with element formulation 14 (axisymmetric) have the normal facing positive Z-direction. All nodes need to be placed with Z = 0 and X 0.

The results of this convergence study can be seen in the Section 5.1.1.

4.5 Validation of the model

Two reports with speciﬁed test setup and data were used to compare patterns of curves as a result of the experiments.

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During the validation of the model, the models were simulated using massively parallel processing (MPP). This type of processing separates the models and solves one part each which leads to a decreased simulation time.

4.5.1 MMALE-method compared to the FOI experiment

The Swedish Defense Research Agency(FOI) has in collaboration with both the Netherlands and Canadian Defense Research Agencies executed simulations and validation tests in one of FOI’s facilities at Grindsjön[28, 29]. The validation tests contained three explosions of 40 gram C4 explosive material before their gauges were destroyed. A thick steel plate was placed at a water depth of 5 meters, with an explosion center varied between 30 to 70 cm above the plate, named A1, A2 and A3. A more detailed description is stated in the FOIs test report[28].

The keywords used and created during the convergence studies in Section 4.2 were implemented for a MMALE- simulation model. The geometry of the simulation model can be seen in Figure 11, where all the lengths are deﬁned in Table 7.

Figure 11: A drawing of the simulation model with the distances as variables. The data of the variables can be seen in Table 7

An illustration of the created model used for this thesis as comparison to the FOI report is shown in Figure 12.

The model includes a part of water, air and a thick shell plate. It was created as a quarter, with symmetry to decrease the simulation time but without losing any accuracy. The surfaces representing the walls were locked for all degrees of freedom (DOF). The mesh consisted of a ﬁne mesh at the closest area of interest and then expanded further away from the detonation center. The element size closest to the detonation area was set to 11 mm and gradually expanding further away from the interesting area.

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Figure 12: A geometry of the simulation model to compare results to FOI. The blue color visualizes the water and the red color is the air.

Table 7: The data for the variables for the FOI simulation model. The variable Lp varied as can be seen Table 9

Ld Lh Lp Lu Lw La Rc

5m 5.3m vary 1.8m 1.5m 0.7m 0.4m

The simulation created in the report speciﬁed the data used for the explosive material C4 and has also been used for the simulation model and is stated in Table 8.

Table 8: Equation of state data for the explosive material C4 using JWL for the FOI validation tests

JWL A 371 GPa B 3.23 GPa R1 4.15 R2 0.95

! 0.3

E0 7 GPa

V0 1

The detonation was performed three times at diﬀerent depths with the same detonation weight of 40g. The cases are named A1, A2 and A3 and are deﬁned within Table 9.

Table 9: The three experiments performed by FOI Water depth [m] Distance to plate Lp[m]

A1 4.7 0.3

A2 4.49 0.51

A3 4.23 0.77

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The experiment had pressure gauges placed on top of the thick plate at a distance of 5cm from the plate’s center. This was replicated when building the simulation model to receive a similar pressure graph.

4.5.2 MMALE-method compared to Australian experiment

The Australian experiment refers to a report made by the Australian Department of defense [33]. The experiment consisted of detonating 250g Pentolite against a 12 meter long cylinder with several strain gauges inside the cylinder. The cylinder was located at a 5 meters water depth and the sea ﬂoor varied from a depth of 12 to 16 meters to the surface. To represent the experiment a model was created using geometry called ”Butterﬂy”

mesh. The geometry and dimensions of the model can be seen in Figure 13 and with the variables speciﬁed in Table 10.

Figure 13: A drawing of the simulation model for the Australian experiment with the variables data in Table 10

Table 10: The data for the geometry in Figure 13 Lq Ls Ld Le Ll Lc Lw Dc

2.5m 2.5m 3m 3m 1.8m 12m 15m 0.4m

The element size closest to the explosion and the interested area was set to 50 mm, with a total of 3 · 10⁶solid ALE elements of the mesh.

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Figure 14: The mesh used for the simulation. The water is colored in blue and the ambient water boundary in red.

The detonation center is marked as a yellow dot in Figure 15

Figure 15: A transparent ﬁgure of the simulation mesh where the cylinder is visible. The little yellow dot is the detonation center.

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Figure 16: The meshed cylinder isolated from the water

The cylinder’s geometry is simplified with no external details and can be seen in Figure 16. The cylinder used for the experiment had extra weight applied at the sides to of the cylinder decrease the buoyancy in combination with some extra steel attachment but have been simplified for the simulation model to just add a weight. The data of the cylinder steel density (C350 with the standard AS1163) and extra weight at both sides of the cylinder by using *Element_Mass_Part are defined in Table 11.

Table 11: The data used for the cylinder visualized in Figure 16.

Cylinder density extra added weight per side

1500 kg/m³ 326 kg

The cylinder is deﬁned with a total of 38594 shell elements of type 16 (fully integrated). The strain gauges are deﬁned as output of the closest shell element that represents the marked location. More details about the Australian experiment can be found in the report[33].

4.5.3 An analytical model of the simulation

In addition to the developed MMALE method described above, another analytical method is used to calculate the strains of the cylinder of the Australian experiment to validate if it is accurate enough to the MMALE model. This analytical method is described in Section 2.5, with the keyword denoted *Load_SSA, which is dependent on several material constants.

The algorithm applies the calculated load at a shell object representing the normal of the element in four different ways. This is ordered to the code using *Set_Part_Column by assigning the first column A1 to the preferred circumstances given in Table 12. The four ways can either be air, the fluid or that the part is neglected by the shock wave.

Table 12: The four ways to represent an object with the analytical method. The part could either be assigned with a ﬂuid or air. The fourth option makes the part neglecting the detonation.

Outside Inside A1 Fluid Fluid 1 Fluid Air 2 Air Fluid 3

Non Non 4

The only geometry used in the analytical model simulation to recreate the Australian experiment is the same cylinder as in the MMALE-model, and is visualized in Figure 16. The algorithm was set to have a distance of 1.8 meters from the detonation to the cylinder and a distance to the surface of 5 meters. The distance to the

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sea bottom is set to 7 meter from the cylinder’s position. The cylinder is assigned to have ﬂuid on the outside and air on the inside, as the normal is pointing outwards in this case, A1 =2. The explosion material is the same as in the experiment, 250 grams of Pentolite. The constants used to describe Pentolite in the analytical model is stated in Table 13.

Table 13: The material constants used to describe the explosive material Pentolite for the Aus

tralian experiment using the analytical method.

A [-] ↵ [-] I [-] K✓ [-]  [-]

5.621e7 1.194 -0.257 8.6e-5 1.4

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5 Results

This section presents the results of the investigated areas. Several convergence studies are presented of both the bubble pulsation and the shock wave peak pressure.

In addition to the mesh studies, Equation of states (EOS) and validation of the patterns from the two reports are presented.

5.1 Convergence studies of bubble pulsation and shock wave

Multiple studies have been made of both the shock wave and the pulsation. During these studies, a spherical TNT explosive detonation has taken place at the center with a radius of 2 cm which equals a weight of 54 grams. The same spherical model in Figure 9 was used during all these simulations, but the element size and amount of elements varied. The bubble pulsation has been simulated during a time frame of 200 ms and the shock wave was simulated for a time frame of only 2 ms of time.

5.1.1 Convergence study of bubble

The inner part closest to the detonation was assigned with a finer mesh in addition to a coarser mesh outside the inner mesh. The number of elements varied from 1000 elements to 550000 elements. With an increased amount of elements it is obvious that the simulation time will increase, as more elements demand a larger matrix of nodes to solve. In Table 14 all the element sizes used for the convergence study are clarified together with the simulation time for both Equation of states of Gruneisen and Linear Polynomial. The results show that the simulation time did not vary using different EOS for a low amount of elements but as the elements increase the simulation time increase at a higher rate for the EOS Linear polynomial.

Table 14: A summary of the element sizes of the convergence in combination with the total number of elements and simulation time.

Convergence run 1 2 3 4 5 6

Amount of elements 1000 11668 52020 78615 400000 550000

Inside element size [mm] 150 75 37.5 37.5 15 15

Outside element size [mm] 450 225 225 112.5 225 112.5

Time of simulation EOS

Gruneisen 12s 2min 38s 28min 35s 45min 47s 12hours 57min 52s 16hours Time of simulation EOS

Linear polynomial 12s 2min 29s 27min 45s 45min 13s 13hours 12min 6s 20hours

As for the mesh size compared to the bubble radius in Figure 17 it can be seen that an increased number of elements implies closer results to the theoretical value. Both EOS are resulting in the same results except for the second convergence run where the Gruneisen gives the exact theoretical radius. This is probably just a coincidence, if the trend line is considered. The graphs of both the radius and the ﬁrst bubble pulse can be seen to result in the same value even though the outer element size decreases. This indicates that there is no need to have an increased mesh outside of the focused area. It will still result in the same value with a shorter simulation time.

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Figure 17: Results of the convergence study showing how the accuracy is improved by increasing the mesh size. The inner element size is the same for the marked points.

In Figure 18, the results of an increased mesh ﬁrst decrease from a large positive error to a negative error. The pattern is similar to the bubble radius in Figure 17, where the outer element size appears to not change the result.

Figure 18: Results of the convergence study of how the mesh is dependent of the accuracy of the theoretical time the bubble reaches its ﬁrst minimum. The inner element size is the same for the marked points.

A series of time steps of the bubbles movement can be seen in Figure 19. Four pictures visualizing the bubble formation; one at the start of the explosion, one when the bubble has reached the maximum radius, one when the bubble is minimized at its ﬁrst minimum and on showing the migration towards the surface after the minimization phase. This is all shown in the same order as for the theory in Figure 5

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(a) After 2 ms of the detonation. The gas bubble is growing

(b) After 42 ms of the explosion. At this time the bubble has reached the ﬁrst maximum ra

dius.

(c) After 82 ms of the explosion the bubble has reached the ﬁrst minimum.

(d) After 114 ms of the detonation. Here it can be seen that the bubble has migrated towards the virtual surface.

Figure 19: The figure shows time steps of an explosion of the gas bubble’s movement at four different times. The yellow elements of the figures are illustrating the water and the brown is the gas bubble. Note that Figure 19d is not the second maximum radius, the bubble is within its expansion phase

5.1.2 Convergence study of pressure shock wave

The shock wave requires a larger amount of smaller elements and has therefore been simulated in a separate study with a shorter time interval of 2 ms. The study has been performed of both an axisymmetric 2D model and a quarter of a 3D spherical model, both are ways to represent the same explosion case. The 2D model is created as a plane of the geometry Figure 9 and the 3D models are similar to the meshes used in the convergence study of the bubble. The pressure is measured at a distance of 0.25 meters from the detonation center.

In Figure 20 several simulations have been studied to investigate how the element size impacts the accuracy of the peak pressure according to the theoretical value given in Equation (5). The total amount of elements during study have varied from 1350 elements to 431250 elements. The amount of elements and simulation time to simulate 2 ms for the 2D axisymmetric study is described in Table 15

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Table 15: A summary of the simulations of the pressure where both the number of elements used for the axisymmetric simulation and the simulation time is given.

Convergence run 1 2 3 4 5 6

Inner element size 50 mm 25 mm 12.5 mm 7.5 mm 3.75 mm 2 mm Amount of elements 1350 5400 16800 40000 128000 431250 Time of simulation 2s 4s 13s 40s 4min 20s 24min 1s

Figure 20: A comparison of pressure wave development for six simulations with diﬀerent element sizes. The red line is visualizing the analytical value calculated with equation (5).

Figure 20 has been focused to show how the curves match the analytical curve and this can be seen in Figure 21. This ﬁgure shows that the size of the mesh makes a big impact of the result of the pressure wave. It can be noticed that when the Analytical result reaches the fraction ^P^max_e which for this case is 31 MPa, the analytical curve slowly starts to diﬀer from the simulated curves. The analytical curve decreases faster towards zero pressure than the simulated pressure curve. This is expected according to the theory in Section 2.1.3.

Figure 21: A focused ﬁgure of the peak pressure for just a tenth of a millisecond of Figure 20.

The black dotted line indicates where the analytical equation is starting to decrease at a higher rate than the reality.

The 2D axisymmetric model was also simulated during a time period of 200 ms to illustrate the pressure response of three bubble pulses. The pressure response generated from convergence run 4 is shown in Figure 22.

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Figure 22: The pressure response of convergence run 4 using 40000 elements. The graph shows three pressure peaks created by the bubble pulsation.

In the same way as the 2D axisymmetric case, a similar study has been made of a 3D model of a quarter of a sphere with a 20 mm radius of a TNT charge. This sphere has varied from 23000 elements to 4400000 elements.

The mesh size, number of elements and simulation time for all the models are shown in Table 16. The actual simulation results are shown in Figure 23.

Table 16: A summary of the simulations of the pressure where both the number of elements and element size used for the 3D-quarter simulation combined with the simulation time is given.

Convergence run 1 2 3 4

Inner element size 50 mm 25 mm 12.5 mm 7.5 mm Amount of elements 23000 178000 1080000 4400000 Time of simulation 1min 7min 1hour 5hr 15min

In Figure 23 all four convergence runs of the 3D spherical mesh named in Table 16 can be visualized in combination with the analytical red line calculated from Equation (5).

Figure 23: A comparison of pressure wave development for four simulations with diﬀerent meshes in 3D to ﬁnd a comparison between element size and accuracy. The red line is visualizing the analytical value calculated with equation (5).

The same curves visualized in Figure 23 above can be seen in Figure 24 of a focused area of just 0.15 ms. Here

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it can clearly be seen that an increased amount of elements increases the accuracy of the pressure of the 3D mesh. The breaking point where the analytical equation starts to decay at a higher rate matches the expected value of 31 MPa.

Figure 24: A focused ﬁgure of the peak pressure for just a tenth of a millisecond of Figure 23.

The black dotted line indicates where the analytical equation is starting to decrease at a higher rate than the reality.

The expansion of the shock wave can be seen in Figure 25. When the shock wave reaches the position where the element’s size increases some numerical reflections appear. These are artifacts of the non-perfect mesh. At first the pressure wave grows clean with no reflections but as it reaches the boundary where the element size increases in Figure 25c it can be seen that some reflections back to the center are starting to develop.

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(a) After 100 µs of the detona (b) After 450 µs of the explosion. (c) After 600 µs of the explosion.

tion. The shock wave is propagating The shock wave reaches 1.5m and The shock wave can be seen to through the water. the shift to an increased element still propagate outwards for the in-

size. creased elements. Some pressure re

ﬂections can be visualized inwards.

Figure 25: The sequence of the shock wave propagation of the 2D axisymmetric simulation also shows reﬂections in Figure 25c from the changing mesh size as an example to show the reﬂections as the element size increases.

5.2 Validation

In addition to the convergence studies showing that a finer inner mesh gives more accurate solutions closer to the analytical equations, a validation of the FSI coupling was considered. The validation of the model consisted of a re-creation of specified tests of external reports that included graphs and a specified setup. Two validation tests were included. One to validate the pressure close to the thick plate and one to measure the strain of a cylinder under water.

5.2.1 FOI

FOI has used several diﬀerent ways to protect the pressure gauges during the experiment while measuring the pressure at the platform below the detonation. One of the graphs has been used as a comparison, named P05 in the report and was placed 5 cm from the center of the thick plate. The results can be seen in Figure 26 of a comparison between experimental data from the report and simulation model data. There are three tests named A1, A2 and A3, all with varied distance to the thick plate as described in Table 9.

It can be seen that the pattern of the red and green curves do have a similar pressure pattern. The red curve has similar peaks as for the experimental but does not reach the same amplitude. For the green curve, it can be seen that the experiment is peaking for a short time but for the simulation it is for a longer time. However the blue curve is not replicating the curve of the experiment. In all three cases the amplitude of the curve is not reached. One reason why the blue curve is not similar could be because of a phenomena called water-jet effect occurring to the plate. This water-jet effect has not been considered during the creation of the model and therefore could have some unknown influence on the pressure gauge. Even though only 40 grams of C4 was used, the highest water-jet velocity during the ”A2” was measured to 170 m/s for the simulation.

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(a) The empirical graph from the FOI experiment [29] (b) The simulation graph represents the pressure 5 cm from of pressure gauge P05 placed 5 cm from the thick plates the center.

center.

Figure 26: Comparison between empirical data and the simulation for the pressure gauge P05.

Three diﬀerent detonation distances were investigated, where A1 was the closest one and A3 was the furthest.

5.2.2 Australian experiment

The Australian experiment presented results of two graphs from what the Australian report refers to as "Event 1" with a distance of 1.8 meter from the explosion center to the cylinder.

The curves in Figure 27a are from the very ﬁrst structural response where the shock wave hits the cylinder . It is from the Australian experiment with comments of the diﬀerent phases of the structural response. The graph in Figure 27b is the strain of the simulation of the same time interval. The red curve is the strain of the axial direction measured on the inside of the shortest distance to the detonation. The blue curve is measured at the same area as the red curve but along the circumferential direction. The green curve of the 45 degree strain (S3) has not been calculated in the simulation and is therefore not included in the graph.

The two graphs are not that similar. The shock wave impact occurs at the same time but then a cavitation response takes place in the experiment, but this cannot be seen in the simulation graph. However, the blue curve shows a similar pattern after the yellow marked area. The same could be said for the red curve even though the high frequencies are not included. The yellow marked cavitation area could according to the report be of the cavitation formation and collapse closest to the cylinder. An explanation could be that this is a result of using a too large element size of the ALE-elements illustrating the water.

(a) The empirical graph from the Australian experi

ment[33]

(b) The result from the simulation of the shock wave

Figure 27: A comparison of the external empirical data with the simulation model at the center of the cylinder.

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In Figure 29 the structural response can be seen of the same explosion at 1.8 meters distance. The strain gauges were placed at a distance of 2.8 meters from the center of the cylinder along the axial direction, all four strain gauges were placed with 90 degrees between. The positions referred to in Figure 29 is illustrated in Figure 28, thus "Port" is the measuring position closest to the detonation and "starboard" is the furthest.

Figure 28: An illustration of the names used in the Australian report, used in Figure 29.

The experimental graph in Figure 29a is noted by the Australian experiment author, with a steel shock wave reaching the gauges ﬁrst, before the water shock wave reaches the same point. This is because the speed of sound is much faster in steel than water. The interferes area marked in yellow is according to the author of the Australian experiment disturbance in the measurement cables, as all gauges received the high frequencies regardless where the gauges were at the cylinder. In the simulation graph, the curves in Figure 29b are named after the name given in the empirical graph. The Y-axis describes the strain.

The simulation curves can be seen to match the curves of the empirical data with the blue curve highest and the green lowest. However, the amplitude of the curves from the simulation are smaller and the high frequencies are not seen in the simulations. If the assumption that the cables were disturbed during 3.5 to 4.7 ms is correct, then the slope of the simulation is probably more correctly showing the strains.

(a) The experimental graph from the Australian experiment [33]

(b) The result from the simulation of the shock wave.

Figure 29: A comparison of the external empirical data with the simulation model at a 2.8 meters distance from the center of the cylinder.

5.2.3 Analytical method applied to the Australian experiment

The analytical method is applied to compare it to both the Australian experiment and the MMALE-model.

In Figure 30, it is shown that the *Load_SSA simulation shows a reasonable agreement to the Australian experiment. The shock wave reaches the cylinder at 1.1 ms, which is the same for the experiment. The blue curve, representing the circumferential direction, shows signs of a large negative strain but during a larger period of time than the experiment. The strain is thereafter continuously altering around zero strain. The axial stain, visualized in red does not represent the cavitation response of the high amplitude, but is 1.4 ms representing the pattern with some offset of the y-axis. The axial strain(red) is rather well predicted except for some high frequency cavitation response initially and that the final oscillation seems to have a negative offset in the experiment.

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(a) The experimental graph from the Australian experi

ment [33]

(b) The analytical method *Load_SSA

Figure 30: A comparison of the external empirical data with the analytical model *Load_SSA at the center of the cylinder.

In the same way as in the ﬁgure above it is shown in Figure 31 that the shock wave reaches the cylinder in the same time as the experiment at approximately 1.6 ms. The ﬁrst 2-3 ms could be considered to be a way to represent the cylinders response with the analytical method but as the simulation continues the strain shows an oscillating behavior not seen in the experiment. This could be a result as the cylinder is not being in water and thereby excluding the water dynamics.

As mentioned in Section 5.2.2, the yellow marked area in Figure 31a is probably some measuring error in the experiment and therefore not commented in this ﬁgure.

(a) The experimental graph from the Australian experi

ment [33]

(b) The analytical method *Load_SSA

Figure 31: A comparison of the external empirical data with the analytical model *Load_SSA at a 2.8 meters distance from the center of the cylinder.

Structural Analysis of Underwater Detonations