Verification and Validation of Modeling of Fluid–Solid Interaction in Explosion-Resistant Designs Using Material Point Method

Abstract

Verifying and validating explosion-resistant design models are challenging tasks due to the difficulties in accurately capturing the failure evolution within a setup influenced by the combined effects of fluid–solid interactions (FSI), blast waves, fragmentation, and impact. Curtain wall system, as a key structural component, is widely used in various types of buildings for its aesthetic appeal and weather protection. Hence, optimizing the explosion-resistance of such systems is necessary to improve building safety. In this work, we develop computational procedures that can be used to enhance the design of blast-resistant structures. This paper focuses on studying a representative component (e.g., window panels) from a typical curtain wall system, as well as a small-scale modeling of shock tube testing. For that, the material point method (MPM) simulations are verified against the finite element method (FEM) simulations, and the computational results are validated against shock tube testing. The work objective is to evaluate the simulation fidelity of explosion responses in several case studies. The first case study demonstrates how the MPM captures damage and fragmentation in a typical confined explosion event involving FSI, thus, providing an improved physical description compared to the FEM. The second case study qualitatively compares the MPM’s ability to simulate the shock tube response with experimental observations. Since the second study validates that the MPM solution is qualitatively consistent with the experimental data, the MPM model is then used in the third case study to establish an FEM model that could capture the same physics. This FEM model can be scaled up to model field experiments. The fourth case study involves the development of an FEM model for a representative curtain wall system component, which is validated against experimental results and then scaled down and employed to validate a corresponding MPM model. The proposed procedure provides a feasible approach to verifying and validating explosion-resistant designs for more general cases.

1. Introduction

A curtain wall system is a combination of metal support members, which are usually made of aluminum or steel, one or more panes of glass, polycarbonate of uniform or variable sizes attached to the support members, and rubber gaskets and silicone. Support members, called mullions, are anchored to a building’s frame and floors but are not structural and do not support the weight of the building. These systems are commonly used on large buildings and skyscrapers to allow light into the building, increase floor space (as compared to exterior structural walls built on each floor), regulate the interior environment, and provide an aesthetically pleasing exterior. Though these systems work well for their intended use, they are very likely to fail in the event of sudden, high-pressure events such as the purposeful or accidental detonation of explosives. Sharp shattered glass, particularly annealed glass, creates potentially lethal hazards for building occupants. These kinds of extreme events are concerning for the safety of governmental buildings around the world [1,2].

The determination of the protection level of the said systems is normally performed with experimental investigation. While in-lab experiments and field testing are fundamental in studying any phenomenon, they are often expensive and even unattainable. For that reason, extensive research efforts have been made to develop computational models that can offer an alternative to those experiments [3]. For example, Saffarini and coworkers [4,5,6] have used nano-mechanics simulations in porous materials to develop a series of scaling laws and constitutive equations to describe the mechanical properties that are observed experimentally in shock absorbent systems. Such models have predicted material properties that can be validated by experiments [7]. In other words, a computational procedure, once verified and validated against reliable experimental data, could be used to evaluate different non-tested design scenarios. Also, this helps in selecting an optimized design to be further validated against a field test.

Despite the need for developing high-fidelity computational tools, there are several challenges in the process. Among those is the modeling of failure evolution within a setup influenced by fluid–solid interactions (FSI). FSI is a complicated physical process and can be found in a wide range of important engineering applications such as explosions, impacts, and underground penetrations. The difficulty arises from the challenging task of coupling the computational solid dynamics domain (CSD) with the computational fluid dynamics (CFD) domain, which, in turn, makes it challenging to capture the failure evolution due to the synergistic effects of blast wave and fragmentation [8,9]. For example, ANSYS software, with the use of the conventional Finite Element Method (FEM), is one of the commonly used physics-based computer codes, and it has been adopted in numerical simulations of blast-resistant designs [10,11]. However, the FEM has some limitations that make it dependent on the quality and quantity of model calibration data. One limitation is that the FEM lacks efficiency and accuracy in capturing damage evolution associated with FSI in a typical explosion event due to serious challenges in the excessive mesh distortions and the treatment of the CFD-CSD interface. Furthermore, the parameters of failure criteria need to be calibrated so that the localization zone size and orientation effects on the FEM solutions could be properly assessed to ensure an objective evaluation of energy dissipation. This is because mesh-based methods, such as FEM, with the use of local elastoplasticity and/or damage models, are limited in evaluating the post-peak structural responses involving interfaces [12]. Therefore, it is imperative to overcome such challenges and develop effective and objective procedures for evaluating explosion-resistant design scenarios.

Several modeling procedures have been proposed to tackle the above challenges. Among those are the Arbitrary Lagrangian–Eulerian (ALE) method [13,14,15], Coupled FEM and Boundary Element Method (BEM) [16], uncoupled CSD-CFD methods [17,18], and Material Point Method (MPM) [19,20]. In the ALE method, the fluid phase and solid phase are simulated using Eulerian and Lagrangian frameworks, respectively [2,14,21,22]. Many studies have successfully simulated the FSI problems, modeling vital structural components like glass panels and RC columns using this approach [2,14]. The Coupled FEM-BEM method is used to model the solid phase using the FEM and the fluid phase using the BEM, and the respective computations are then combined to describe the overall system response [23,24]. The uncoupled CSD-CFD methods, such as those employed by Baum [17] and commercial AUTODYN-2D & 3D [18], are used to simulate weapon detonation/fragmentation and dynamic failure resulting from blast and fragment interaction with a target. Baum [17] uses the “master-surface” technique for dealing with multi-phase interactions along with a fast interpolation algorithm to describe loads, displacements, and velocities. The AUTODYN commercial software, which is widely employed in such applications, employs three interconnected solution schemes to deal with the FSI. It employs a Lagrange scheme to model solids, an Euler scheme to model fluids, and an ALE method-based scheme to provide automatic rezoning of distorted grids.

Although AUTODYN is widely used in engineering applications, it is limited in simulating the FSI involving failure evolution due to the uncoupled CFD and CSD treatment. For example, CFD can be used to model the explosive detonation, shock wave propagation, and reflected pressure on a structure, and then a CSD simulation can model the structural response under a given pressure–time history, which is applied to the structure through a nodeset. In addition, the problem of large mesh distortion in the Lagrangian framework (CSD domain) and the issue of fixed mesh connectivity matrix pose significant limitations on the use of mesh-based computer codes when real-time multi-phase failure evolution, such as impact, penetration, and perforation, is simulated.

Meshfree particle methods are representative of advanced spatial discretization methods for circumventing the difficulties inherent in the mesh-based methods when dealing with complicated problems involving failure evolution and multi-phase (solid–liquid–gas) interactions. As a continuum-based particle method, the MPM consists of hybrid Lagrangian–Eulerian solution steps for taking advantage of both solution schemes. It has demonstrated great potential in evaluating extreme loading events [19]. However, the computational expenses of MPM are higher than those of FEM so that the FEM and MPM could be combined into a single computational domain for the balance between accuracy and efficiency in large-scale simulations. In other words, the MPM is used in the part that requires detailed information, while the FEM is employed in the other parts of the whole system [19].

Since the verification of a computational procedure can be performed by comparing two or more corresponding numerical results, it is necessary to find the high-fidelity result for comparison. Without supercomputing facilities, however, it is not feasible to adopt the MPM code for large-scale coupled CSD-CFD simulations. Hence, this work presents the MPM results for key structural components at smaller scales, which are compared against the FEM results and validated against in-lab experiments. As a result, the FEM system model could then be better established with suitable element sizes and mesh orientations for the objective evaluation of energy dissipation in the post-peak regime. In other words, the main objective of this paper is to propose and demonstrate a verification and validation procedure for better evaluating explosion-resistant design scenarios without invoking supercomputing capability.

The following sections of this paper are organized so that the computational procedure is described in Section 2. Section 3 presents the case studies designed to verify and validate the computational results. Section 4 summarizes the major findings and concluding remarks.

2. Computational Procedure

The MPM was originally motivated by the need for better simulations of impact and penetration problems at the macroscale [20,25]. The MPM is an extension of the Particle in Cell (PIC) method that was a modified version of the Fluid Implicit Particle (FLIP) method for CFD [26,27]. Since the early 1990s, the MPM has been applied to many areas in simulation-based engineering science [20,25,28,29,30,31], with recent advances made for improved multi-physics (wave-diffusion-steady state) simulations, multi-phase simulations, and high-order accurate and smooth discretization [32,33,34,35]. The essential idea of the MPM is to take advantage of both the Eulerian and Lagrangian discretization methods while avoiding the shortcomings of each so that it effectively integrates CFD with CSD in a single computational domain for high-fidelity simulations of multi-phase interactions involving damage evolution.

Figure 1 demonstrates an explicit MPM calculation cycle in a single timestep. The method discretizes a continuum body into a set of finite Lagrangian particles (material points) with each particle representing a sub-continuum body of finite mass that carries all state variables at any given time step so that the crushed inertial effect could be well simulated on the subsequent impact response of the layered composite system, avoiding element erosion as needed in the FEM. This discretized system is superimposed over an Eulerian background grid that is used to solve (differentiating and integrating) the governing equation. At every time step, the information carried on the particles is mapped to the background grid to perform all nodal calculations and then remapped again to the corresponding particles to update their positions, velocities, stresses, and all other changes in state variables. Through these continuous mapping and remapping operations, the MPM takes advantage of both Lagrangian and Eulerian descriptions. Such a coupled approach in a single computational domain does not require master/slave nodal treatment for the problems involving impact and/or multi-phase interaction and/or failure evolution. In other words, the use of “single-valued” mapping operations between particles and background grid results in a “natural no-slip contact/impact treatment” at the interface between different material phases and avoids any “inter-penetration”. However, the computational cost of MPM is about twice that of FEM due to the mapping operations between material points and corresponding background grid nodes [19,25]. Since the MPM is based on the same weak formulation as that for the FEM, the MPM and FEM have been combined into a single computational domain for large-scale simulations with the balance between accuracy and efficiency [19].

2.1. Governing Equations

The governing equations can be formulated based on the conservation laws, kinematic conditions, and constitutive models. For isothermal cases considered here, the conservation laws of mass and linear momentum imply that of energy. In direct notation, the conservation of mass and momentum in a continuum body can be described as follows:

$${\displaystyle \frac{d\rho }{dt}}+\rho \nabla \cdot \mathit{v}=0,$$

(1)

$$\rho \mathit{a}=\nabla \cdot \mathit{\sigma }+\rho \mathit{b},$$

(2)

where $\rho (\mathit{x},t)$ is mass density, $\mathit{v}(\mathit{x},t)$ is velocity vector, $\mathit{a}(\mathit{x},t)$ is acceleration vector, $\mathit{\sigma }(\mathit{x},t)$ is Cauchy stress tensor, and $\mathit{b}(\mathit{x},t)$ is specific body force. It should be noted that in this paper, unless noted otherwise, tensors of first rank or higher are denoted by bold-faced letters.

Since the MPM adopts a Lagrangian description to track the material points as deformation occurs, the conservation of mass is inherited by the default solution scheme. As for linear momentum, the weak formulation of Equation (2) becomes the following:

$${\int }_{\Omega }\rho \mathit{a}\cdot \mathit{w}d\Omega =-{\int }_{\Omega }\mathit{\sigma }:\nabla \mathit{w}d\Omega +{\int }_{{\Gamma }_{\sigma }}\overline{\mathit{\tau }}\cdot \mathit{w}d\Gamma +{\int }_{\Omega }\rho \mathit{b}\cdot \mathit{w}d\Omega ,$$

(3)

where $\mathit{w}$ is a test function, $\overline{\mathit{\tau }}$ is a traction vector, $\Omega$ is a continuum body, and ${\Gamma }_{\sigma }$ is a part of the boundary with a prescribed traction. The test function $\mathit{w}$ is assumed to be zero on any boundary with a prescribed displacement.

2.2. Discretization Scheme

The mass density of a continuum body that is discretized into a set of $N_p$ material points in an MPM algorithm can be expressed as follows:

$$\rho \left(\mathit{x},t\right)={\sum }_{p=1}^{N_p}{M}_p\delta (\mathit{x}-{\mathit{x}}_p^t),$$

(4)

where $\delta$ is the Dirac delta function with a dimension of volume inverse. It must be noted that in all equations from this section onwards, the subscript $i$ denotes grid node values, whereas subscript $p$ denotes material point values. Substituting Equation (4) into Equation (3) yields the following:

$$\begin{array}{l}{\sum }_{p=1}^{N_p}{M}_p\left[\mathit{w}\left({\mathit{x}}_p^t\right)\cdot \mathit{a}\left({\mathit{x}}_p^t\right)\right]\\ ={\sum }_{p=1}^{N_p}\left\{-\mathit{\sigma }\left({\mathit{x}}_p^t\right):\nabla \mathit{w}{|}_{{\mathit{x}}_{\mathit{p}}^{\mathit{t}}}+M_p\left[\mathit{w}\left({\mathit{x}}_p^t\right)\cdot \mathit{b}\left({\mathit{x}}_p^t\right)\right]\right\}+{\int }_{{\Gamma }_{\sigma }}\mathit{w}\left({\mathit{x}}_p^t\right)\cdot \overline{\mathit{\tau }}d\Gamma ,\end{array}$$

(5)

By utilizing shape functions, either nodal shape functions as in the original MPM or particle-based shape functions as in the Generalized Interpolation Material Point (GIMP) method to eliminate cell-cross errors [19], Equation (5) can be rewritten to describe Newton’s equations of motion on the background grid nodes as follows:

$$m_i^t{\mathit{a}}_i^t=({\mathit{f}}_i^t{)}^{\mathrm{i}\mathrm{n}\mathrm{t}}+({\mathit{f}}_i^t{)}^{\mathrm{e}\mathrm{x}\mathrm{t}},$$

(6)

where $m_i^t$ is the lumped mass, $({\mathit{f}}_i^t{)}^{\mathrm{i}\mathrm{n}\mathrm{t}}$ is the internal force vector, and $({\mathit{f}}_i^t{)}^{\mathrm{e}\mathrm{x}\mathrm{t}}$ is the external force vector of the grid node $i$, and they are defined as:

$$m_i^t={\sum }_{p=1}^{N_p}{M}_p{N}_i({\mathit{x}}_p^t),$$

(7)

$$({\mathit{f}}_i^t{)}^{\mathrm{int}}=-{\sum }_{p=1}^{N_p}{\displaystyle \frac{M_p}{{\rho }_p^t}}{\mathit{\sigma }}_p^t:\nabla N_i({\mathit{x}}_p^t),$$

(8)

$$({\mathit{f}}_i^t{)}^{\mathrm{e}\mathrm{x}\mathrm{t}}={\sum }_{p=1}^{N_p}{M}_p{N}_i\left({\mathit{x}}_p^t\right){\mathit{b}}_p+{\int }_{{\Gamma }_{\sigma }}{N}_i({\mathit{x}}_p^t)\overline{\mathit{\tau }}d\Gamma ,$$

(9)

in which $N_i\left({\mathit{x}}_p^t\right)$ is the shape function and $\nabla N_i\left({\mathit{x}}_p^t\right)$ is the corresponding gradient associated with node $i$. Each shape function with its corresponding gradient is evaluated at the material point.

2.3. Constitutive Equations

2.3.1. Fluids

For compressible fluids, the relation between the stress tensor and strain rate is given by:

$$\mathit{\sigma }=2\mu \dot{\mathit{\epsilon }}+\lambda \mathrm{t}\mathrm{r}\left(\dot{\mathit{\epsilon }}\right)\mathit{I}-\widehat{p}\mathit{I},$$

(10)

where $\mu$ is the shear viscosity of the fluid, $\lambda$ is the bulk viscosity of the fluid, I is the second-order identity tensor, $\widehat{p}$ is the fluid pressure, $\mathit{\sigma }$ is the stress tensor of the fluid material point, and $\dot{\mathit{\epsilon }}$ is the strain rate tensor of the fluid material points. For the case of gases due to explosion, which will be the only type of fluid in this work, the shear and bulk viscosities terms can be ignored so that Equation (10) takes the following form:

$$\mathit{\sigma }=-\widehat{p}\mathit{I}.$$

(11)

The fluid pressure $\hat{p}$ can be determined using Equation (12):

$$\widehat{p}=\left(\gamma -1\right)\rho e+q$$

(12)

where $q$ is the artificial viscosity, e is the specific internal energy, $\gamma$ is the ratio of specifc heats, and $\beta$ is the bulk modulus. For the case of EOS, the density and specific internal energy for each material point are updated every timestep as follows:

$${\rho }^{k+1}={\displaystyle \frac{{\rho }^k}{1+tr(∆{\epsilon }^{k+1})}}$$

(13)

$$e^{k+1}=e^k+{\displaystyle \frac{({\sigma }^{k+1}:∆{\dot{\epsilon }}^{k+1})}{{\rho }^{k+1}}}$$

(14)

The artificial viscosity (q) is determined by Equation (15), and it is added to the pressure in EOS to eliminate the numerical noises of the shock front, as follows:

$$q=\left\{\begin{array}{r}\rho L_c\left[c_o{L}_c{\left(tr\left(\dot{\epsilon }\right)\right)}^2-c_{1}atr\left(\dot{\epsilon }\right)\right], tr\left(\dot{\epsilon }\right)<0\\ 0, tr\left(\dot{\epsilon }\right)>0\end{array}\right.$$

(15)

where a is the local sound speed of the fluid, $c_o$ and $c_1$ are user-defined constants, and $L_c$ is the characteristic length calculated by $L_c=\sqrt[3]{V_{cell}}$, where $V_{cell}$ is the volume of the grid cell.

2.3.2. Explosives

The explosives are modeled as compressible fluids whose pressure is calculated by Jones–Wilkins–Lee (JWL) EOS. The JWL EOS takes the following form:

$$\widehat{p}=C_1\left(1-{\displaystyle \frac{\omega }{r_{1}V}}\right)\exp \left(-r_{1}V\right)+C_2\left(1-{\displaystyle \frac{\omega }{r_{2}V}}\right)\exp \left(-r_{2}V\right)+{\displaystyle \frac{\omega \psi }{V}}$$

(16)

where $C_1$, $C_2$, $r_1$, $r_2$, and $\omega$ are constants, $V={\displaystyle \frac{{\rho }_o}{\rho }}$ the relative volume with ${\rho }_o$ and $\rho$ being the initial and current density, respectively, and $\psi$ denotes the internal energy.

2.3.3. Solids

To obtain the stress for a solid material point, an elastoplasticity model with strain hardening in a rate form is used as follows:

$${\mathit{\sigma }}^{\nabla }=\mathit{D}:\dot{\mathit{\epsilon }},$$

(17)

where $\dot{\mathit{\epsilon }}(\mathit{x},t)$ is the strain rate tensor, $\mathit{D}$ is the elastoplastic tangent stiffness tensor, and ${\mathit{\sigma }}^{\nabla }$ is the Jaumann stress rate. The use of the Jaumann stress rate ensures that the obtained stress tensor is objective to spin and rotation [19,37,38]. It is imperative to calculate the strain increment $\mathsf{\Delta }\mathit{\epsilon }$ at each time step as part of the process to perform any stress-related calculations for the material point. Based on the previous work [19], the strain increment can be found as follows:

$$\begin{array}{l}\mathsf{\Delta }{\mathit{\epsilon }}_p^{k+1}=\mathsf{\Delta }t\cdot {\dot{\mathit{\epsilon }}}_p^{k+1}\\ =\mathsf{\Delta }t\cdot {\displaystyle \frac{1}{2}}{\sum }_{i=1}^{N_n}\left[{\mathit{v}}_i^{k+1}\nabla N_i\left({\mathit{x}}_p^k\right)+{\left({\mathit{v}}_i^{k+1}{\nabla N}_i\left({\mathit{x}}_p^k\right)\right)}^T\right]\end{array}$$

(18)

The corresponding stress increment could then be obtained via constitutive modeling and an appropriate integration scheme in the strain-stress space [39,40]. Hereafter, two different integration schemes are implemented in the MPM algorithm and then tested using a simple collision (impact) problem to evaluate the balance between accuracy and efficiency.

Single-Step Integration Scheme

This integration scheme is based on obtaining an incremental stress value with the corresponding incremental strain value from Equation (18). For that, a well-known assumption is used so that an incremental strain can be decomposed into elastic and plastic incremental strains so that $\mathit{d}\mathit{\epsilon }={\mathit{d}\mathit{\epsilon }}^{\mathit{e}}+{\mathit{d}\mathit{\epsilon }}^{\mathit{p}}$. In addition to performing the simulations in increments of very small timesteps (in the order of ${10}^{-7}$), the Jaumann stress rate is used to account for rotations that are accompanied by large deformations. The combination of solving the problem in incremental form and the use of the Jaumann stress rate will ensure increments of small deformations at each timestep, allowing the use of the assumption of additive decomposition to the strain increment ($\mathit{d}\mathit{\epsilon }$). Accordingly, the incremental stress $d\sigma$ can be calculated using a tangent modulus (${\mathit{D}}_{\mathit{t}\mathit{a}\mathit{n}}$) and the incremental strain so that $\mathit{d}\mathit{\sigma }={\mathit{D}}_{\mathit{t}\mathit{a}\mathit{n}}\mathit{d}\mathit{\epsilon }$ [35,36]. The tangent stiffness tensor depends on the information at the end of the previous time step so that an integration error will be accumulated in the single-step integration scheme for nonlinear stress–strain relations.

Iterative Newton–Raphson Integration Scheme

Another approach to determining the stress is to use the concept of yield surface in the well-established elastoplasticity theory. This allows the involvement of predefined internal variables that are used to track the stress history during material yielding and deformation. Generally, the yield surface can be expressed as a function of the stress and internal variables so that $f\left(\sigma ,I_i\right)=0$, where I = 1, 2, 3, …, and it does not represent a grid node as suggested earlier. The internal variables $I_i$ can be a tensor of any order to represent the material constituents. To avoid explaining the fundamentals of the classical theory of elastoplasticity, the formulation in this section will be briefly presented without in-depth details. For more details on the general formulation of the theory, the reader is referred to [40].

Given an isotropic yield surface and an associated flow rule in this work, the elastoplasticity model is integrated with an iterative process. When yielding occurs, the iterative process is initiated using the Newton–Raphson method until a predefined tolerance is achieved. The step-by-step integration scheme is outlined in a flowchart as shown in Figure 2 and is described as follows:

• In each time step, the calculations are performed only if the state of stress, total strain, elastic strain, plastic strain, and internal variables from the previous time step ($t-1$) are available. Accordingly, in the first iterative loop ($k$ = 0) within a specific time step ($∆t$), the current state is determined so that ${\epsilon }_k^i={\epsilon }_{t-1}^i$ and ${{(I}_i)}_k={{(I}_i)}_{t-1}$. If the iteration loop $k$ > 0, then the stress is obtained so that ${\sigma }_k=E:({\epsilon }_t-{\epsilon }_k^i)$, in which the superscript denotes the inelastic strain.
• Check if yielding occurs by evaluating the yield surface. If yielding does not occur, then the solution is converged, and the stress value is the one obtained in the previous step. If yielding occurs, then the flow rule and incremental stress need to be evaluated.
• Using the Newton–Raphson method, determine the sub-increment of the $\lambda$ parameter, that is ${\delta \lambda }_k$ so that ${\delta \lambda }_k={\displaystyle \raisebox{1ex}{${-f}_k$}\!\left/ \!\raisebox{-1ex}{${\left(\frac{df}{d\lambda }\right)}_k$}\right.}$, where ${\left({\displaystyle \frac{df}{d\lambda }}\right)}_k=\left({\displaystyle \frac{df}{d{I}_i}}{E}_{pl}\right)$.
• The flow rule and updated plastic modulus are evaluated in this step, as detailed in Ref. [40].
• The internal variables and current state of the strain are determined as ${\left(I_i\right)}_{k+1}={\left(I_i\right)}_{k+1}+{\delta \lambda }_k{h}_k$, ${\epsilon }_{k+1}^i={\epsilon }_k^i+{\delta \lambda }_k{M}_k$.
• The calculation is repeated until the solution is converged for the next time step.

It is important to note that such an iterative integration scheme is sensitive to the chosen time stepping. For large-scale simulations, the balance between efficiency and accuracy must be carefully considered if the computer codes are not parallelized and if supercomputing facilities are not available. This paper is designed to show the MPM capability, with an in-house three-dimensional MPM code without parallelization, in overcoming the challenges of simulating the FSI with the FEM. Hence, only small-scale problems are used in the case studies.

Evaluation of Integration Schemes

To evaluate the above two integration schemes, a collision problem is considered to show the effects of both integration schemes on the results. The designed problem is a collision between two copper blocks both traveling toward each other at a speed of 100 m/s. Figure 3 shows the problem setup of the two impacting copper blocks, where each is 1 ft long and 2 inches in height.

Seven simulations were performed using this setup, four of which employed the same constitutive model but different integration schemes with varying numbers of iterations. The other three simulations used three different constitutive models with the same integration scheme. The three models used are the Bilinear elastoplastic von Mises model (BLVM), Nonlinear elastoplastic von Mises Model (NVM), and the Johnson–Cook Model (JC). The BLVM utilizes a constant tangent modulus to describe the strain hardening. The NVM uses a nonlinear description of strain hardening with an associated flow rule [40] so that the yield function, flow rule, and internal state variable take the forms of

$$f\left(\sigma ,I_1\right)=3J_2-H^2\left(I_1\right)$$

(19)

$$M\left(\sigma ,I_1\right)=N\left(\sigma ,I_1\right)={\displaystyle \frac{1}{N}}{\displaystyle \frac{\partial f}{\partial \sigma }}$$

(20)

$$dI=d\lambda$$

(21)

where H is a hardening function of the internal variable $I$, and is assumed as follows:

$$H=\left\{\begin{array}{cc}{H}_o+\left(H_L-H_o\right) sin\left[{{\displaystyle \frac{\pi }{2}}\left({\displaystyle \frac{I}{I_L}}\right)}^n\right],\hfill & \hfill 0\le I<I_L\\ H_L ,& \hfill I\ge I_L\end{array}\right.$$

(22)

where $H_o$ is the yield strength, $H_L$ is the ultimate strength, $I$ is the plastic strain invariant, and $n$ is a constant calibrated from the uniaxial tensile test.

The JC model, as shown by Equation (23), is used as well to verify that the code is irrelevant to the type of the constitutive law utilized here. For that, the model ignores the terms related to strain rate and temperature by setting them equal to 1. This is to make sure that the model produces a fit like that obtained by the NVM model, as shown in Figure 4, which displays the uniaxial stress–plastic strain response of the three constitutive models used here. In other words, the problem is designed so that the NVM and the JC models have almost identical stress–strain fits to minimize the effects of strain rate and temperature and to keep the results of the more advanced models (i.e., JC and NVM) comparable to the simplest one (BLVM).

$${\sigma }_{JC}=\left(A+B{{\epsilon }_p}^n\right)\left(1+Cln\left({\displaystyle \frac{\dot{\epsilon }}{{\dot{\epsilon }}_o}}\right)\right)\left(1-{\left({\displaystyle \frac{T-T_r}{T_m-T_r}}\right)}^m\right)$$

(23)

In Equation (23), ${\epsilon }_p$ is the plastic strain, $\dot{\epsilon }$ is the strain rate, ${\dot{\epsilon }}_o$ is the reference strain rate, T is temperature, $T_m$ is melting temperature, $T_r$ is the reference temperature, and A, B, C, n, and m are material constants. The values of material constants as obtained from the literature for copper are A = 90 MPa, B = 292 MPa, and n = 0.31 [41].

The simulation results are summarized here through the measured particle velocity at the free surface of either block. Figure 5 shows the particle velocity measured at the free surface of one of the copper blocks for the seven different simulations. The results clearly show that the three models have insignificant differences in capturing the velocity at the free surface (orange, blue, and black curves). However, the effect of the numerical integration scheme can be seen clearly. This is because the incremental–iterative integration scheme could reduce the stress error with the increase in iteration numbers. Although there is a time difference among different numbers of iterations, the differences in particle velocity amplitude among different integration schemes are about 15%, with consistent profiles for all the simulations. Hence, the single-step integration scheme will be adopted for the case studies to keep the balance between accuracy and efficiency in simulations.

For all the case studies in the next section, the explicit MPM algorithm [19,20] will be used with the single-step integration scheme in the constitutive space [40]. Much research has been conducted to eliminate the cell-crossing error in the original MPM, for which the GIMP is the first attempt in that direction [19,29]. However, all the advanced versions of MPM would increase computational expenses for simulation accuracy. Hence, in this work, we will utilize the MPM solution procedure with an ad hoc approach [19,25], in which the velocity vector of any material point very close to the cell boundary is fixed without updating during the corresponding time step. With the use of such an explicit time integration procedure that requires very small timesteps to meet the stability criterion, the material point will not cross more than one cell within a single time step. As a result, the balance between accuracy and efficiency could be obtained in the following case studies.

3. Results (Case Studies)

In this section, several numerical examples are presented for the case studies that are designed to achieve four main objectives. The objectives are addressed here and summarized in

Table 1. Overall objectives of the case studies considered in this paper.

Case Study ID	Overall Objective	Designed Problem
1	Show the MPM strength in capturing FSI involving damage evolution as compared with the FEM.	Concrete box containing explosives
2	Build an MPM model of the shock tube test and validate it qualitatively against the experiment.	Shock tube test
3	Scale Up (Bottom-Up) approach Establish an FEM model using the validated MPM model from the second case study to simulate an actual shock tube test.	Shock tube test
4	Scale Down (Top-Down) Approach Develop an FEM model for a problem that does not involve FSI or fragmentation. The model is then scaled down to be used as a basis to verify the corresponding MPM model.	Laminated Glass Panel subjected to blast.

:

• The first case study is designed to show how the MPM captures fragmentation and damage in a typical confined explosion event, which also demonstrates the comparison between the FEM and MPM in simulating the FSI.
• The second case study is designed to compare, qualitatively, the MPM’s ability to capture the shock tube response against an actual shock tube test.
• Since the second case study validates the MPM solution to be qualitatively consistent with the experiment, the MPM model will be used in the third case study to build an FEM model that captures the physics as simulated by the MPM. This FEM model is needed to be scaled up to model large-scale experiments whenever feasible.
• The fourth case study involves the development of an FEM model that is validated against experimental results. This FEM model is then scaled down and used to validate a corresponding MPM model. Such an approach enables the integration of MPM with FEM in a single computational domain for evaluating large-scale explosion-resistant scenarios.

3.1. First Case Study

As explained earlier, the FEM is limited in the quantitative evaluation of FSI involving failure evolution due to its use of uncoupled CSD-CFD computational domains and element erosion. This case study illustrates the MPM’s ability to capture fragmentation and FSI resulting from explosion scenarios, as compared with the uncoupled CSD-CFD solver in AUTODYN, based on the previous work [28].

3.1.1. Computational Model Setup

The problem is a plane strain condition of a hollow concrete box containing TNT explosive materials inside of it. The concrete box has 2 m side length containing a box of explosives that has 150 mm side length. The same exact setup is adopted for both the MPM and FEM simulations. Figure 6 shows the MPM and the FEM model setup and geometry.

The FEM model was built using a hybrid 2D uncoupled Lagrangian–Eulerian multi-material solution scheme as built-in AUTODYN and Explicit Dynamics Solvers in the ANSYS WORKBENCH. As a combined CFD-CSD scheme, the FEM Lagrangian mesh for the solid concrete box is built using Explicit Dynamics Solver and then imported into AUTODYN to be overlapped on an Eulerian mesh that will cover the air and explosives flow. The mesh generated by the explicit solver for the solid was adopted without any modifications by the user.

3.1.2. Constitutive Laws

Concrete Box

Both the MPM and FEM utilize a bilinear elastoplasticity model with an equivalent plastic-strain-based damage criterion for concrete. This constitutive model is the simplest model to simulate the typical uniaxial stress–strain curve of concrete as shown in Figure 7.

Table 2. Concrete elastoplasticity model parameters.

Density (kg/m3)	Young’s Modulus (GPa)	Tangent Modulus (GPa)	Poisson’s Ratio	Yield Strength (MPa)
2300	25	6.25	0.18	15

shows the values of the model parameters appearing in Figure 7.

To further simplify the simulation, the softening portion is ignored, and material failure is assumed to occur at the point of ultimate strength. The strain value at which the failure occurs is set to be 0.3% (0.003). It should be noted that when a failure strain value is achieved, the particle stress is zeroed, and it is kept at that level for the remainder of the solution time. When the particle has a zero-stress value, it remains visually apparent due to the particle mass during the post-processing process because no element-erosion is needed in the MPM.

Ambient Air

The ambient air is modeled as a compressible fluid whose stress is determined as shown in Section 2.3.1.

Table 3. Air parameters used for modeling ambient air.

Density (kg/m3)	1.225
Specific Heat Constant Pressure (J/ kg K)	1006.4

shows the model parameters used for air in the simulation.

TNT Explosives

The TNT explosives are modeled as compressible fluids whose pressure is calculated by Jones–Wilkins–Lee (JWL) equation of state (EOS), as shown in Section 2.3.2.

Table 4. JWL parameters used for modeling TNT explosives.

${\mathit{C}}_1$ (GPa)	${\mathit{C}}_2$ (GPa)	${\mathit{r}}_1$	${\mathit{r}}_2$	$\mathit{\omega }$	${\mathit{\psi }}_{\mathit{o}}$ (MJ/m3)	${\mathit{\rho }}_{\mathit{o}}$ (kg/m3)
373.8	3.747	4.15	0.9	0.35	6000	1630

shows the values used for the constants in the JWL EOS.

3.1.3. Results

Figure 8a shows the literature result [28], Figure 8b shows the MPM solution, and Figure 8c shows the FEM solution. Although a more advanced constitutive model (elastoplasticity with decohesion) was used in the literature model [28], the MPM solution with a simple perfect-plasticity constitutive model is reasonably close to the literature results. The two figures do not show an exact match, but the discrepancy can be easily explained by the effect of the different constitutive laws used in both simulations. Also, the damage in our MPM simulations has been oversimplified compared to the damage criterion followed in the literature model [28]. In our simulations, we utilize a simple element-erosion-like technique. It is similar in concept (because it eliminates the contribution of the damage particle) but different in application (because it does not follow the same methodology in visually representing the damage). For particles that have failed, they are represented as “damaged” within the algorithm. This means that the particle has entered a state of zero stress and will not contribute to the material strength or state of stress. The particle’s information will not have any stress carried over once it is damaged. This will prevent it from contributing to the solution on the grid nodes in a similar manner to as if it were removed using the element erosion approach.

For the FEM solution, however, the explosive pattern and concrete box failure pattern are not consistent with the physics or the MPM solution despite using the same constitutive law as the MPM solution. This can be attributed to the use of an uncoupled CFD-CSD approach. This case study illustrates that the MPM solution can physically capture explosion scenarios with better accuracy than the FEM, even with the use of a very simple and approximated material model. Also, the MPM is more accurate in simulating the explosive and FSI behaviors as compared with the FEM.

3.2. Second Case Study

The objective of this case study is to simulate the shock tube test using the MPM due to its strength in capturing FSI and failure evolution, as shown in the first case study. A qualitative assessment means that the same setup typically found in a shock tube test is adopted but on a smaller scale. The overall response is expected to be the same despite the dimensions being different. It is beyond the scope of this paper to discuss the actual shock tube test which is detailed in a previous work [42]. Here, one of the calibration tests performed in the actual shock tube experiment is used as a reference to perform the qualitative assessment.

3.2.1. Overview of the Experiment

A high explosives-driven shock tube was used to create a desired pressure versus time waveform on a steel plate. A steel plate that is 38 inches wide by 66 inches tall by 0.25 inches thick was mounted at the end of the shock tube and secured with a 3.5-inch “bite” using 0.5-inch-thick bearing plates and jam screws around the perimeter on 6-inch centers, as shown in Figure 9. Holes were drilled and tapped to accept reflected pressure sensors on the vertical and horizontal centerlines of the steel plate, with additional holes in the support structure just outside the steel plate. This allowed for variable placement of sensors from test to test based on desired measurement locations. A laser was positioned to measure the deflection at the center of the steel plate. The piezoelectric reflected pressure sensors (PCB Model 102B18) are capable of measuring at 1 MHz, and the deflection laser (Acuity Model AR700-50) is capable of measuring at 5 kHz. The testing procedure and data collection followed along with the descriptions found in Ref. [42]. For this specific test, seven reflected pressure sensors along with one deflection laser were used to fill the eight-channel data acquisition system.

3.2.2. Computational Model Setup

Figure 10 shows the MPM model setup. This setup simulates the actual shock tube setup [42] but uses a smaller size. In the actual test, the explosives are usually placed at a distance of 55 ft, and the tested steel plate is 5.5 ft in height and 0.25 inches in thickness.

3.2.3. Constitutive Laws

The constitutive laws utilized here for the air and the explosives are the same as those discussed in Section 3.1.2. The shock tube is considered as a fixed, rigid body that does not deform. However, structural steel is used to model the tube. As for the steel plate, it is simulated using the material model discussed next.

Steel Plate

The strain rate and temperature-dependent Johnson–Cook (J-C) material model are integrated into the MPM code to better predict the objective energy dissipation. The JC model adopted for the steel material is given in Equation (23). The values of material constants as obtained from the literature for Steel 1006 are A = 350 MPa, B = 275 MPa, n = 0.36, ${\dot{\epsilon }}_o=1$, C = 0.022, m = 1, $T_m$ = 1811 °C, $T_r=25$ °C, and $T=25$ °C. Strain rate ($\dot{\epsilon }$) is calculated at each time step as $\dot{\epsilon }={\displaystyle \frac{\left(\frac{{\sigma }_{eq}-{\sigma }_{yield}}{1.5G+E_p}\right)}{{\dot{\epsilon }}_o/dt}}$, where ${\sigma }_{eq}$ is the equivalent stress from the previous timestep, ${\sigma }_{yield}$ is yield strength, $G$ is shear modulus, $E_p$ is the plastic modulus, $dt$ is the timestep.

3.2.4. Results

Figure 11 shows the results obtained from the MPM solution and experiments. As shown in Section 3.2.2, the MPM model is built with dimensions that are different from those in the actual experiment. In addition, the explosive types and properties as well as the distance at which the detonation occurs in the MPM model, do not represent the actual experiment. For those reasons, comparing exact numbers between the model and the experiment is ignored, and a qualitative assessment is pursued here. The objective is to assess the response through the behavior of the deflection measure at the center of the steel plate. The experiment shows that the deflection at the plate center takes a sinusoidal pattern that is attenuated with time. There are six peaks, as highlighted by small green circles, and an almost plateau response, as highlighted by a larger green circle. The first peak represents the maximum deflection in the plate center upon the arrival of the shock wave. The remaining peaks represent the steel plate vibrational response after resisting the initial shock wave. These peaks get attenuated as time progresses until the vibration fades out. There is no damage or failure in the steel plate.

3.3. Third Case Study

In this case study, the same shock tube MPM model is used to build an equivalent model using the FEM scheme as used in the first case study. The objective is to tune the FEM scheme using the MPM such that the FEM is able to capture the important characteristics. In other words, since the previous two case studies were used to show that the MPM can capture the important aspects of the physics involved in a typical explosion event, it is important to fine-tune the corresponding FEM model so that it can be later Scaled Up (Bottom-Up approach) to perform the actual large-scale simulations.

3.3.1. Computational Model Setup

The MPM model setup is detailed in Section 3.2.2. Here, the FEM model is detailed. The FEM model is developed using a hybrid Lagrangian and 2D Eulerian multi-material scheme in AUTODYN. As a combined CFD and CSD scheme, the FEM mesh is built using LS-DYNA and then imported into AUTODYN to be imposed over an Eulerian mesh to simulate the air and explosives. Figure 12 shows the problem setup.

3.3.2. Constitutive Laws

The same material models used in the MPM model from the second case study were used for the FEM model. The reader is referred to Section 3.2.3 for more details.

3.3.3. Results

Figure 13 illustrates the deflection measured at the center of the plate for both FEM and MPM models. The simulation results show that the FEM model was successful in capturing the MPM results in a quantitively consistent manner.

The explosion occurs at the center of the simulated tube (as shown in Figure 12) at the beginning of the simulation time ($t_{simulation}$ = 0 s). However, the first moment of notable deformation in the steel plate occurs a few milliseconds after the simulation starts. The zero time point in Figure 12 ($t_{figure}$ = 0 s) corresponds to the moment at which the first notable deformation occurs in the FEM model (This would correspond to $t_{simulation}=0.03ms$). Everything earlier than that moment in time is ignored due to its irrelevance to the current study. The corresponding moment is also considered for the MPM plot.

Figure 14 shows the shock front around the explosives and through the fluid while pushing against the steel plate. It can be seen that the deformation patterns are significantly similar for the two models.

The findings from this case study show that for simulating a shock tube test, the MPM and FEM solutions can both capture the essential features of explosion scenarios if the FEM model is developed by utilizing the MPM solution. In fact, the small-scale MPM is used to calibrate major components of the FEM model, such as the mesh size, the FSI interaction scheme, and the constitutive model, by adjusting the FEM software parameters. This overcomes the FEM limitations resulting from the CSD-CFD uncoupling. Using the supercomputing capability, the coupled FSI with the MPM could better predict the explosive responses than the uncoupled FSI in the FEM codes. In particular, the objective simulation results could be obtained with the MPM-based constitutive modeling if the size effect on failure evolution is calibrated against well-designed blast/impact experiments for precision engineering, which is beyond the scope of this paper.

3.4. Fourth Case Study

The following case study is designed to further illustrate the ability of MPM to model failure evolution without considering the FSI. Thus, the blast load can be modeled using the pressure history extracted from the experiment. For that purpose, this problem is chosen to compare the MPM solution to an already-developed FEM model to further test the MPM’s ability to capture the failure response. In fact, this benchmark problem could enable the integration of the strengths of MPM with FEM in a single computational domain to balance computational accuracy and efficiency. The FEM is a powerful approach for large-scale simulations, while the MPM is effective in capturing large-deformation-related multi-physics with its coupled CSD-CFD computational schemes.

3.4.1. Overview of the Experimental Test Setup

It is beyond the scope of this paper to discuss the details of the experimental work because they can be found in the reference by Idriss et al. [43], and the data were shared with the authors by the U.S. Army Protective Design Center. However, this section will provide a brief overview of the test setup that was used to validate the FEM model results presented in the coming sections.

Table 5. Field data of window system.

Sample	Dimensions
Outer ply (blast side)	0.125 in. annealed
Interlayer	0.06 in, PVB
Inner ply	0.125 in. annealed
Measured average blast loading
Pressure (psi)	6.6

shows the window system layer thicknesses that were tested and then used in the FEM model. The laminated glass system was made of a single unit that is 60 in. long × 48 in. wide. The window consisted of two 1/8-in.-thick annealed panes and a 0.06-in.-thick PVB interlayer. In this study, one side (inner ply in the experiment) receives the blast load. As detailed in Ref. [43], the field experiments on laminated glass systems were performed by clamping the panel edges to a rigid steel frame with a rubber strip that was 1 in. wide on one face and 0.5 in. wide on the other face of the window. The panel’s loaded area was 56 in. × 44 in. The peak average pressure used in the field test was 6.6 psi.

3.4.2. Computational Model Setup

There are two different FEM models developed for this case study. The first is a large-scale FEM model that was developed to model the actual field test presented in the previous section. The second model is the same as the first one but scaled down to a smaller size that can be simulated by the MPM. The aim is to use the FEM model, which is developed to capture the experimental response, as a reference to verify the MPM model results. By scaling down (Top-Down Approach) the first model, it becomes possible to compare the MPM model with experiments indirectly.

Large-Scale FEM Model

The FEM model was developed using LS-DYNA. Figure 15 shows the window panel after building the geometry in ANSYS Design Modeler following the details of the field test setup presented in Section 3.4.1. The system is composed of two glass layers, one polymer interlayer, and rubber gaskets that are typically placed between the glass and the steel window frame to provide fixation. The top surface of both gaskets is fixed in all directions. The laminated glass plies are modeled as layers within a single composite body. Multiple through-thickness mesh elements were used to account for multiple layers of glass and polymer, as shown in the figure. The laminated glass system has the same dimensions as those presented in Section 3.4.1.

Small-Scale FEM Model

Figure 16 shows the small-scale model of the same laminated glass system shown earlier, except that it is scaled down to a smaller size. The window system is made of a single unit that is 12 in. long × 12 in. wide. The window consists of two 1/8-in.-thick annealed panes and a 0.06-in.-thick PVB interlayer. This is the same setup adopted after the large-scale model but for a smaller window size. The panel edges are simulated as if clamped to a 1 in. wide bite on one face and a 0.5 in. wide bite on the other face of the window, similar to what has been adopted in the large-scale model. The panel’s loaded area was 10 in. × 10 in.

MPM Model

Figure 17 shows the geometry of one-quarter of the window system presented in Figure 16, which is discretized into material points to be simulated using the MPM code. The boundary conditions were applied at the grid nodes that are closest to the material points that fall within the bite area. Panel (b) of Figure 17 shows (in red) the material points that fall in the bite region. The grid nodes that are closest to those material points are the nodes that are assigned with fixed boundary conditions to mimic the gasket fixation shown in the FEM model. To further reduce the computational cost, a quarter of the window was simulated while maintaining symmetry along the two in-plane perpendicular directions.

3.4.3. Constitutive Laws

Glass

An elasto-damage model (isotropic elasticity with the maximum tensile strain failure criterion) was used for the glass material with a peak strength of 6000 psi, elastic modulus of 10⁷ psi, and a Poisson’s ratio of 0.22 [44]. A prescribed threshold failure strain value is used to trigger failure of glass.

Polymeric Interlayer

To model the dynamic response of the polyvinyl butyral (PVB) interlayer material, the material is assumed to stretch plastically without failure using the ANSYS built-in Johnson–Cook (JC) material model as described in Equation (23). The only exception is that the thermal part of the model is assumed to be the one to ignore the thermal effect on inelastic response. The values of material constants as obtained from the fits are A = 2 psi, B = 10 psi, n = 0.3, and C = 0.25. The JC model parameters listed here were extracted from the results of uniaxial dynamic tensile tests obtained from Ref. [45]. The experimentally extracted stress–strain relation is shown in Figure 18, along with the fitted JC model. The interlayer mechanical properties are shown in

Table 6. Material Properties of PVB Component.

Property	Value
Density (lb/in3)	0.0397
Young’s modulus, psi	27,400
Poisson ratio	0.495
Elastic limit, psi	2400
Failure strain	1.20

.

Even though the fitted JC model appears to approximate the experimental response with a “perfect-plasticity-like” behavior, it was chosen because it has a strain rate-dependent term. Also, it is common to model the dynamic response of PVB interlayers using a JC model rather than a perfect plasticity model, as shown in the previous literature [46].

3.4.4. Results

Validation of the Large-Scale FEM Model Against Experiments

The deflection-time plot of the experimental results and the FEM model results are compared in Figure 19. The predicted deflections by the FEM model are in good agreement with the field data. The average reflected peak pressure of the blast measured in the field experiment was 6.6 psi, which is higher than the maximum pressure needed to break the annealed glass pane in tension during plate bending. Therefore, it was expected that the annealed glass pane would break during the first few milliseconds of the blast event. Maximum deflections differed by 4.17% between the field test and the FEM model estimates.

Verification of the MPM Model against the Scaled-Down FEM Model

Figure 20 shows the comparison between the results of the MPM model and the FEM model. The preliminary results shown are up to 15 ms out of the full 20 ms needed to reach the peak deflection value reported by the experiment and the 35 ms needed to reach the complete destruction of the window. For the time beyond 15 ms, the MPM model requires computational resources that are currently not available at our facilities. Throughout the simulation, significant agreement between the two models is clearly noticed.

The accuracy of the MPM model, as compared with the FEM results, provides a successful preliminary verification of the MPM model. To further verify the MPM model, Figure 21 shows the deformation contours after a few milliseconds from the FEM simulations (Panel a) and the MPM simulations (Panel b). The figure clearly shows that the MPM model, qualitatively, captures the same deformation pattern that the FEM model is showing. In other words, we can see the zero displacement at the fixed supports, the localized displacement at the corner, and the same scale color change along the plot diagonal line.

4. Summary and Conclusions

Curtain wall systems are important in modern architecture. However, they are vulnerable to failure during high-pressure events such as explosions and hurricanes, which create hazards for building occupants. With experimental validation, computational methods could be used to design and evaluate different scenarios for optimizing the design of blast-resistant curtain wall systems to mitigate the effect on the glass, curtainwall frame, and structural components of the building. Conventional finite element modeling in ANSYS software is commonly used in numerical simulations of blast-resistant designs, but there exists a great need for improvement due to the limitation of mesh-based methods in describing fluid–solid interaction and failure evaluation with objective energy dissipation. The MPM is an advanced spatial discretization method that can be used to circumvent difficulties inherent in mesh-based methods when dealing with complicated problems that involve failure evolution and multi-phase (solid–liquid–gas) interactions. However, due to the large-scale size of the structural components in a typical curtain wall system, it may not be feasible to adopt in-house MPM codes for coupled computational fluid and solid dynamics simulations of the whole system response with the available supercomputing facilities. In this paper, an in-house MPM code has been used for small-scale validation and verification against the FEM results and experiments for several case studies. In the first case study, the MPM demonstrated its ability to capture FSI, fragmentation, and damage in a typical confined explosion event as compared to FEM. The second case study is designed to compare, qualitatively, the MPM’s ability to capture the response of a simplified shock tube test as compared with the actual shock tube test. The developed MPM model is physical and couples the CFD and the CSD domains. The validated MPM solution is then used in the third case study to build a physical and small-scale FEM model of the same problem. This FEM model can now be scaled up to model actual shock tube experiments if needed, making it possible to compare the numerical results quantitatively with the experimental data. The fourth case study involved developing an FEM model, which is verified against experimental results, and then the FEM model is scaled down and used to verify a corresponding MPM model.

The proposed scheme in this paper aims to provide a balanced approach for researchers with limited access to supercomputing facilities and experimental setups. By utilizing the advantages of each computational technique along with a predesigned verification and validation scheme, researchers can develop needed models to simulate real life blast scenarios. Such simulations will eventually advance the design of structural systems subjected to blast loadings. The reason for the specific selection of MPM and FEM to be the computational schemes to approach the problem lies in the ability of each method to fill the two main gaps in simulating FSI problems. Those two gaps are the lack of coupling between CFD and CSD domains and the simulation of large-scale problems. MPM was originally developed to simulate large deformation FSI problems, but it is unable to simulate large-scale problems. On the other hand, FEM is a widely used method to simulate large-scale problems and actual experimental setups but lacks the easy implementation of coupled CFD-CSD analysis. Combining the two methods makes them the ideal candidates for achieving our objective of proposing and demonstrating a verification and validation procedure for better evaluating explosion-resistant design scenarios without invoking significant supercomputing capability. However, the findings from this paper support the need for investing in further efforts that combine the MPM and FEM in a single computational domain to integrate the strengths of both methods while overcoming the limitations of each. As a result, system-wide computational models could then be verified and validated for optimizing the blast-resistant system design. In other words, the results presented in this paper need to be considered with care; even though the approach is promising and has great potential, further development and a more thorough evaluation are needed. Researchers need to be careful when choosing valid comparative schemes while carrying out such an approach. Also, the physics needs to be thoroughly investigated and verified with benchmark problems prior to any large-scale implementation of the procedure.