Volume 3, Issue 4 e1137

RESEARCH ARTICLE

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An adaptive coupling of finite elements with smooth particle hydrodynamics particles for large deformation fluid–solid interactions

Arijit Khan,

Corresponding Author

Arijit Khan

[email protected]

orcid.org/0000-0002-5234-7423

APAC | E&U BU, CGI Information Systems and Management Consultants Pvt Ltd, Bangalore, India

Correspondence Arijit Khan, APAC | E&U BU, CGI Information Systems and Management Consultants Pvt Ltd, 124-125, Divyashree Technopolis, Yemlur Main Road, Off Old Airport Road, Bangalore, Karnataka 560037, India.

Email: [email protected]

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Debasis Deb,

Debasis Deb

Department of Mining Engineering, Indian Institute of Technology, Kharagpur, India

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Arijit Khan,

Corresponding Author

Arijit Khan

[email protected]

orcid.org/0000-0002-5234-7423

APAC | E&U BU, CGI Information Systems and Management Consultants Pvt Ltd, Bangalore, India

Email: [email protected]

Search for more papers by this author

Debasis Deb,

Debasis Deb

Department of Mining Engineering, Indian Institute of Technology, Kharagpur, India

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First published: 18 November 2020

https://doi.org/10.1002/cmm4.1137

Citations: 3

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Abstract

This article presents a simple technique to couple the finite element method (FEM) and the smooth particle hydrodynamics (SPH) method in the Lagrangian framework for the simulation of problems involving solids and structures under shock loading that cause large deformations, such as rock blasting. For such problems FEM suffers from mesh distortion and thus requires some numerical corrective measures such as element erosion or remeshing to proceed with the solution. These cause some artificial effects and a sufficient increase in the computational time. Mesh-less particle based methods such as SPH, on the other hand offer the flexibility to handle high-speed motion inherently. But the SPH method is more computationally demanding and requires special treatment for application of boundary conditions. For the simulation of certain process that involve high impact by fluid on solids or structures such as blasting, a coupled FEM-SPH procedure might be more efficient where regions of large deformations are modeled with SPH particles and regions far off are modeled with FEM. An adaptive procedure of coupling FEM-SPH is developed in this work so that highly distorted elements in the model are replaced by SPH particles on the fly but remain linked to the model thus maintaining the consistency in solution. The article provides the details of this adaptive procedure. Validation of this coupling method is done by analyzing a two-dimensional (2D) elastic impact problem and then is applied for the 2D simulation of fragmentation in brittle rocks induced by high impact blast loads from explosions considering a continuum damage model under plain strain conditions. Since detailed three-dimensional (3D) FSI simulations require, even for the lone SPH or FEM solver, quite powerful computer resources, most examples in this article are restricted to two dimensions assuming plane symmetry and can easily be extended for 3D analyses.

1 INTRODUCTION

The finite element method (FEM) has long been the tool of choice for engineers and scientists needing to simulate problems governed by partial differential equations since the theory is well developed and robust. One of the shortcoming of the method is that for problems that involve large deformations the mesh gets highly distorted to a point where accuracy is lost and stability of the method may vanish. Over the years several methods are developed such as mesh rezoning, element erosion, and so forth to do away with badly distorted elements. These methods may provide results but suffers from computational irregularity and increase computational cost. Alternatively, modified FEM such as the arbitrary Lagrangian Eulerian method may be used for such problems with the introduction of convective terms in the governing equations.¹

Mesh-less methods on the other hand make an interesting proposition to model such problems since they can handle large deformations inherently.² One such particle-based method is The smooth particle hydrodynamics (SPH) by References 3 and 4 which has been used extensively to model explosions (free and confined) and problems under high impact loading conditions.^{5, 6} SPH method was applied to simulate crack propagation⁷ and recently for blast induced damage.⁸ But it is more computationally expensive than FEM¹ and produces some instabilities when applied to problems of deforming solids. Furthermore, application of Neumann type boundary conditions is also a tedious process in this method. It must be mentioned here that the Boundary-Element Method that works by discreatizing the boundary of the domain alone, an advantage that is off-set by the difficulties associated in integrating the Green's function for nonlinear problems.⁹

To get the best of both methods a coupled FEM-SPH technique is proposed where regions of large deformations are modeled with SPH particles to capture deformations and the remaining regions are modeled with FEM to optimize computational time and seamlessly apply these boundary conditions.

In order to couple the FEM and Mesh-free particle methods contact and attachment between elements and particles needs to be established and the transfer of information between them must be coherent. Some of the proposed algorithms in this direction can be found in the works of References 10-14, and more recently in References 15, 16, and 17.

This work has adopted with suitable modifications of the algorithm proposed by Xiao¹⁸ for simulating impact and penetration problems in solids where they treat elements near the coupling interface as imaginary particles to calculate forces on particles from adjacent elements and impose equivalent traction on element edges on the coupling interface in accordance with the particle stress associated with that element. Rock basting is a fluid–solid interaction problem with high impact shock loading followed by expansion of the explosive gas that act on the solid causing large strain near the boundary of the blast hole. Hence it is prudent the explosive material is exclusively modeled with SPH particles. The rock is considered as a Lagrangian material and is solved with FEM under suitable boundary conditions. As the simulation proceeds if any rock element is found to be distorted it is replaced by a equivalent solid SPH particle.

The details of this procedure is given in the following sections and is organized as follows. Section 2 states the basic equations of the SPH method. Section 3 briefly describes the Lagrangian formulation of the FEM. In Section 4, the damage modeled used in this work is presented. In Section 5, the details of the FEM-SPH coupling algorithm is provided and Section 6 presents some numerical example for validation of this method followed by an example of explosion in rocks to which the algorithm is applied.

2 THE SPH METHOD

The SPH method essentially comprises of two fundamental concepts. First, is the integral representation of a function and second is the particles approximation of the problem domain. For through explanation and development one is referred to Reference 19.

2.1 The governing equations

The governing equations of the problem are the basic conservation laws of mass, momentum and energy. The SPH discretized form are:

Conservation of mass

\frac{D ρ_{i}}{D t} = \sum_{j = 1}^{N} m_{j} (v_{i}^{β} - v_{j}^{β}) \frac{\partial W_{i j}}{\partial x_{i}^{β}} .

()

Conservation of momentum

\frac{D v_{i}^{α}}{D t} = \sum_{j = 1}^{N} m_{j} (\frac{σ_{i}^{α β}}{ρ_{i}^{2}} + \frac{σ_{j}^{α β}}{ρ_{j}^{2}} - Π_{i j} δ^{α β}) \frac{\partial W_{i j}}{\partial x_{i}^{β}} .

()

Conservation of energy

\frac{D e_{i}}{D t} = \frac{1}{2} \sum_{j = 1}^{N} m_{j} (\frac{σ_{i}^{α β}}{ρ_{i}^{2}} + \frac{σ_{j}^{α β}}{ρ_{j}^{2}} + Π_{i j} δ^{α β}) (v_{j}^{α} - v_{i}^{α}) \frac{\partial W_{i j}}{\partial x_{i}^{β}},

()

where

ρ

is the density, m is the mass, v is the velocity,

σ

is the stress, e is the specific internal energy per unit mass, x is the coordinate, i and j are particle indices,

α, β

are the Cartesian coordinate indices and

δ

is the Kronecker delta function. W is the kernel function, and

Π

is the artificial viscosity, needed for handling shocks. The summation is taken over the total number of neighbors N of particle i. It is referred to Reference 5 for detailed derivation of Equation (1). The position of any particles is given by:

\frac{D x}{D t} = v .

()

The Kernel function used in this work is the B-Spline function of the form:

W_{i j} = W (d_{i j}, {\overline{h}}_{i j}) = α_{d} \times \{\begin{matrix} 2 / 3 - ξ^{2} + ξ^{3}, & 0 \leq ξ < 1, \\ {2 - ξ}^{3} / 6, & 1 \leq ξ \leq 2, \\ 0, & ξ > 2, \end{matrix}

()

where

ξ

d_{i j} / {\overline{h}}_{i j}, d_{i j} = \sqrt{x_{i j}^{α} x_{i j}^{α}}, x_{i j}^{α} = x_{i}^{α} - x_{j}^{α}, h_{i j} = 0.5 (h_{i} + h_{j})

, h is the smoothing length, and

α_{d}

is a constant dependent on the problem dimension, which is taken as

15 / 7 π h^{2}

for two-dimensional problem. The initial smoothing length is given as

h_{0} = α_{h} Δ

where h₀ is the initial smoothing length,

α_{h}

is a dilation factor and

Δ

is the particle spacing. The smoothing length is updated with

h = h_{0} {(\frac{ρ_{0}}{ρ})}^{1 / 2} .

()

For the artificial viscosity

Π

, the expression suggested by Monaghan²⁰ is used, which is given as

\begin{align} Π_{i j} = \{\begin{matrix} \frac{- α_{Π} {\overline{c}}_{i j} ϕ_{i j} + β_{Π} ϕ_{i j}^{2}}{{\overline{ρ}}_{i j}} & v_{i j}^{α} x_{i j}^{α} < 0, \\ 0, & v_{i j}^{α} x_{i j}^{α} > 0, \end{matrix} \end{align}

()

where

c_{i j} = 0.5 (c_{i} + c_{j}), ϕ_{i j} = \frac{{\overline{h}}_{i j} v_{i j}^{α} x_{i j}^{α}}{{| d_{i j} |}^{2} + ϕ^{2}}, ϕ = 0.1 {\overline{h}}_{i j}, {\overline{ρ}}_{i j} = 0.5 (ρ_{i} + ρ_{j})

, c denotes the speed of sound, and the constants,

α_{Π}

and

β_{Π}

, are both set to 1.0.

SPH suffers from a defect documented in the literature as “Tensile Instability” which make particles clump together. This is treated by introducing an “Artificial Stress” term,

({R_{i}}^{α β} + {R_{j}}^{α β}) {(f_{i j})}^{n}

()

in the momentum equation and the factor f is given by Gray et al.²¹ as:

f_{i j} = \frac{W (d_{i j})}{W (Δ)} .

()

The value of the index n is taken to be 4. The artificial stress in the principle direction is given as:

\begin{align} {\overline{R}}_{i}^{α α} = \{\begin{matrix} - ε \frac{{\overline{σ}}_{α α}}{{\overline{ρ}}^{2}} & {\overline{σ}}_{α α} < 0, \\ 0, & {\overline{σ}}_{α α} \geq 0, \end{matrix} \end{align}

()

where

{\overline{σ}}_{α α}

is the principle stress in the

α

direction and ε is the dispersion constant taken to be equal to 0.1. The term is obtained by coordinate transformation from principle coordinate to the original coordinate system.

2.2 Constitutive relation

The rate-based constitutive law for elastic material is used where the Cauchy stress rate tensor is:

\begin{align} {\dot{σ}}^{α β} = \frac{d σ^{α β}}{d t} . \end{align}

()

The objective stress rate of the Cauchy stress, the Jaumann Stress rate

({\hat{σ}}^{α β})

is used:

\begin{align} {\hat{σ}}^{α β} & = D_{α β κ λ} {\dot{ε}}^{κ λ} \end{align}

()

\begin{align} = \dot{σ^{α β}} + σ^{α γ} ω^{γ β} - ω^{α γ} σ^{γ β}, \end{align}

()

where

D_{α β κ λ}

is the fourth-order elasticity tensor (Stiffness modulii) of the material and the stretch tensor (or rate of deformation tensor) is defined as:

{\dot{ε}}^{α β} = \frac{1}{2} (\frac{\partial v_{α}}{\partial x_{β}} + \frac{\partial v_{β}}{\partial x_{α}})

()

for large displacements the rotation tensor cannot be neglected and is defined as:

ω^{α β} = \frac{1}{2} (\frac{\partial v_{α}}{\partial x_{β}} - \frac{\partial v_{β}}{\partial x_{α}}) .

()

The SPH form of the above equations for any particle i is:

{\dot{ε}}_{i}^{α β} = \frac{1}{2} [\sum_{j = 1}^{N} \frac{m_{j}}{ρ_{j}} (v_{j}^{α} - v_{i}^{α}) \frac{\partial W_{i j}}{\partial x_{i}^{β}} + \sum_{j = 1}^{N} \frac{m_{j}}{ρ_{j}} (v_{j}^{β} - v_{i}^{β}) \frac{\partial W_{i j}}{\partial x_{i}^{α}}],

()

ω_{i}^{α β} = \frac{1}{2} [\sum_{j = 1}^{N} \frac{m_{j}}{ρ_{j}} (v_{j}^{α} - v_{i}^{α}) \frac{\partial W_{i j}}{\partial x_{i}^{β}} - \sum_{j = 1}^{N} \frac{m_{j}}{ρ_{j}} (v_{j}^{β} - v_{i}^{β}) \frac{\partial W_{i j}}{\partial x_{i}^{α}}] .

()

2.3 Time integration in particle method

For the integration in time of the rate Equations (1a)–(1c) and (14)–(15) the second-order accurate leap-frog scheme was used in this work. The reader is referred to Reference 19 for detailed implementation of this scheme. The position is updated by:

x_{t + Δ t} = x_{t - Δ t} + \overline{v} Δ t,

()

where

\overline{v}

is the smoothed velocity derived from velocity of all neighboring particles given by the XSPH formulation²² such that each particle move together with its neighboring particles to make the motion more localized and stable. The expression is given as:

\begin{align} {\overline{v}}_{i, t + Δ / 2}^{α} = v_{i, t + Δ / 2}^{α} + \tilde{e} \sum_{i = 1}^{N} \frac{m_{j}}{ρ_{j}} (v_{j, t + Δ t / 2}^{α} - v_{i, t + Δ t / 2}^{α}) W_{i j}, \end{align}

()

where

\tilde{e}

is a constant called the smoothing constant taken to be 0.5.

2.4 Determination of time step size

Due to the conditional stability of the above explicit time integration scheme, it is imperative that the time step size

Δ t

is chosen according to the Courant–Friedrichs–Lewy (CFL) condition that states that the maximum rate of information travel must not exceed the wave celerity in the material.

Δ t_{CFL} \leq \frac{Δ x}{c},

()

where

Δ x

is the characteristic length in the discretized model and c is the maximum speed of sound in the material. For SPH system,

Δ x = \min_{a} (h_{a})

; h_a implying the smoothing length of particle a. Furthermore, Reference 23 included the effect of artificial viscosity in the calculation of

Δ t

Δ t_{CFL} = \min_{a} (\frac{λ h_{a}}{c_{a} + h_{a} | \nabla . v_{a} | + 1.2 (α_{Π} c_{a} + β_{Π} h_{a} | \nabla . v_{a} |)}) .

()

Equation (19) multiplied by a tuning factor ( $λ = 0.3$ ) was used in this work.

3 LAGRANGIAN FORMULATION FOR LARGE DEFORMATION ANALYSIS

The total Lagrangian formulation is used in this work and the details can be found in Reference 24. Here only a very brief description is given for completeness.

The formulation begins with the deformation gradient F:

\begin{align} F = \frac{\partial x}{\partial ξ}, \end{align}

()

where x is the spatial coordinates which is the position in the current configuration and

ξ

is the material coordinates which is the position in the reference configuration. In addition, if a is the displacement vector then,

x = ξ + a

and in tensor notation:

\begin{align} F_{i j} = δ_{i j} + \frac{\partial a_{i}}{\partial ξ_{j}} . \end{align}

()

The Green–Lagrange strain

Γ

tensor is defined as:

\begin{align} Γ = \frac{1}{2} (F^{T} . F - I) . \end{align}

()

Using Equation (21) in Equation (22) the following is obtained:

\begin{align} Γ_{i j} = \frac{1}{2} (\frac{\partial a_{i}}{\partial ξ_{j}} + \frac{\partial a_{j}}{\partial ξ_{i}}) + \frac{1}{2} (\frac{\partial a_{k}}{\partial ξ_{i}} \frac{\partial a_{k}}{\partial ξ_{j}}) . \end{align}

()

The increment in the Green–Lagrange strain tensor is got by,

Γ^{(t + 1)} - Γ^{t}

and is of the form:

Δ Γ = \frac{1}{2} (F^{T} . \nabla_{0} (Δ a) + {(\nabla_{0} (Δ a))}^{T} . F) + \frac{1}{2} {(\nabla_{0} (Δ a))}^{T} . \nabla_{0} (Δ a) .

()

The work conjugate stress measure used is the second Piola–Kirchhoff stress tensor

τ

, which is related to the Cauchy stress tensor

σ

through:

\begin{align} σ = \frac{ρ}{ρ_{0}} F . τ . F^{T}, \end{align}

()

where the current density

ρ

is given by:

\begin{align} \det F = \frac{ρ_{0}}{ρ} . \end{align}

()

With the strain and stress measures unambiguously defined the principle of virtual work can be invoked with respect to some reference configuration:

\begin{align} \int_{V_{0}} δ Γ^{T} τ^{t + Δ t} d V = \int_{S_{0}} δ a^{T} t_{0} d S + \int_{V_{0}} ρ_{0} δ a^{T} g d V, \end{align}

()

{\vec{t}}_{0}

being the nominal traction on S₀ and

\vec{g}

the gravity vector. Here, the initial undeformed configuration of the system at time t = 0 is chosen as the reference, a formulation called the “Total Lagrangian formulation” is thus obtained.

\begin{align} (K_{L} + K_{N L}) Δ a = K^{\tan} Δ a = f_{ext}^{t + 1} - f_{int}^{t}, \end{align}

()

where

Δ a

is the unknown increments in nodal displacement vector from time t to time

t + Δ t

. The material stiffness contribution K_L is:

\begin{align} K_{L} = \int_{V_{0}} B_{L}^{T} D B_{L} d V \end{align}

()

and the geometric stiffness contribution K_NL is:

\begin{align} K_{N L} = \int_{V_{0}} B_{N L}^{T} ϒ^{t} B_{N L} d V . \end{align}

()

The complete expressions of the linear and nonlinear strain-displacement relation terms B_L and B_NL and the second Piola–Kirchoff stress matrix

ϒ

, as related to Equation (23) can be found in Reference 24. The external force vector and the internal force vector are defined as:

\begin{align} f_{ext}^{t + Δ t} = \int_{S_{0}} H^{T} t_{0} d S + \int_{V_{0}} ρ_{0} H^{T} g d V, \end{align}

()

\begin{align} f_{int}^{t} = \int_{V_{0}} B_{L}^{T} τ^{t} d V, \end{align}

()

where H is the shape function matrix, t₀ is the traction applied in initial configuration on surface S₀, g is acceleration due to gravity, and

ρ_{0}

is the initial specific gravity of the material.

3.1 The generalized– $α$ method

The equation of motion of a material subjected to external perturbations is a modification of Equation (28) to include the inertial term and a viscous damping term as:

M \ddot{a} + C \dot{a} + f_{int}^{t} = f_{ext}^{t} .

()

Herein M is the mass matrix, C is the proportional damping matrix, f^int is the vector of nonlinear restoring force, f^ext is the vector of externally applied load, and

\ddot{a}

\dot{a}

, and a are the vector of acceleration, velocity, and displacements, respectively.²⁵ The complete set of algebraic equations in time is obtained using the finite difference method comprising the generalized-

α

method by Chung and Hulbert.²⁶ In a most general framework the equation of motion is modified and applied to general mid-points

(α_{m}

and

α_{f})

within each time step instead of the end-point as in classical Newmark's scheme. These parameters also control the dissipative characteristics of the algorithm. Thus at the (t + 1)th time-step or at time

t + Δ t

it is of the form:

\begin{align} M {\ddot{a}}^{t + 1 - α_{m}} + C {\dot{a}}^{t + 1 - α_{f}} + f_{int}^{t + 1 - α_{f}} = f_{ext}^{t + 1 - α_{f}} . \end{align}

()

The superscripts denote the time discrete combinations of the quantities defined as:

\begin{align} {\ddot{a}}^{t + 1 - α_{m}} & = (1 - α_{m}) {\ddot{a}}^{t + 1} + α_{m} {\ddot{a}}^{t}, {\dot{a}}^{t + 1 - α_{f}} = (1 - α_{f}) {\dot{a}}^{t + 1} + α_{f} {\dot{a}}^{t} \\ a^{t + 1 - α_{f}} & = (1 - α_{f}) a^{t + 1} + α_{f} a^{t}, f_{ext}^{t + 1 - α_{f}} = (1 - α_{f}) f_{ext}^{t + 1} + α_{f} f_{ext}^{t} \end{align}

and the vector of internal forces f_int that depend nonlinearly on the displacements:

f_{int}^{t + 1 - α_{f}} = (1 - α_{f}) f_{int}^{t + 1} + α_{f} f_{int}^{t} .

()

Using Newmark's approximations²⁷ for (j + 1)th iteration (j = 0, 1, 2 … , n) the time discrete combinations of velocities and accelerations can be written as:

{\dot{a}}_{j + 1}^{t + 1 - α_{f}} = (1 - α_{f}) [(\frac{γ}{β Δ t}) (a_{j + 1}^{t + 1} - a^{t}) - (\frac{γ - 2 β}{2 β}) Δ t {\ddot{a}}^{t}] - (\frac{γ (1 - α_{f}) - β}{β}) {\dot{a}}^{t},

()

{\ddot{a}}_{j + 1}^{t + 1 - α_{m}} = (1 - α_{m}) [(\frac{1}{β Δ t^{2}}) (a_{j + 1}^{t + 1} - a^{t}) - (\frac{1}{β Δ t}) {\dot{a}}^{t}] - (\frac{1 - α_{m} - 2 β}{2 β}) {\ddot{a}}^{t} .

()

Using the above equations, Equation (34) can be expressed as:

\begin{align} M [(1 - α_{m}) [a_{0} (a_{j + 1}^{t + 1} - a^{t}) - a_{2} {\dot{a}}^{t} - a_{3} {\ddot{a}}^{t}] + α_{m} {\ddot{a}}^{t}] + C [(1 - α_{f}) [a_{1} (a_{j + 1}^{t + 1} - a^{t}) - a_{4} {\dot{a}}^{t} - a_{5} {\ddot{a}}^{t}] + α_{f} {\dot{a}}^{t}] \\ + [(1 - α_{f}) (f_{i n t, j + 1}^{t + 1} - f_{ext}^{t + 1}) + α_{f} (f_{int}^{t} - f_{ext}^{t})] \equiv G (a_{j + 1}^{t + 1}) = 0, \end{align}

()

where,

\begin{align} a_{0} & = \frac{1}{β Δ t^{2}} a_{1} = \frac{γ}{β Δ t} a_{2} = \frac{1}{β Δ t} \\ a_{3} & = \frac{1}{2 β} - 1 a_{4} = \frac{γ}{β} - 1 a_{5} = (\frac{γ - 2 β}{β}) \frac{Δ t}{2} . \end{align}

()

First-order Taylor series expansion of Equation (38) results in the effective iterative structural equation as:

\begin{align} {\overline{K}}_{j} (a_{j}^{t + 1}) d a \approx - G (a_{j}^{t + 1}), \end{align}

()

where

{\overline{K}}_{j} (a_{j}^{t + 1})

is the deformation dependent effective stiffness matrix, given by:

{\overline{K}}_{j} (a_{j}^{t + 1}) = \frac{\partial G (a_{j}^{t + 1})}{\partial a^{t + 1}} = (1 - α_{m}) M a_{0} + (1 - α_{f}) C a_{1} + K_{j}^{t + 1 - α_{f}}

()

and da is the iterative increment in displacements from iteration j to j + 1.

K_{j}^{t + 1 - α_{f}}

for the modified N-R scheme is given as

(1 - α_{f}) K^{t}

. A usable relation regarding the choice of Newmark's parameters (

γ, β

) and the mid-points (

α_{m}, α_{f}

) is given by Chung and Hulbert²⁶ as:

\begin{align} α_{m} & \leq α_{f} \leq \frac{1}{2} \\ β & = \frac{{(1 + α_{f} - α_{m})}^{2}}{4} \\ γ & = \frac{1}{2} + α_{f} - α_{m} \end{align}

()

if Equation (42) is satisfied then the algorithm is unconditionally stable and second-order accurate in the linear regime with optimal combination of high-frequency and low-frequency dissipation. The out of balance load vector G is checked to be less than

ϵ_{f} = 1 \times 1 0^{- 4}

for convergence. This formulation along with the time integration scheme will hence forth be called as FEM to distinguish it from standard FEM with fixed grid.

3.2 Shock resolution by artificial viscosity

In the presence of shocks, the governing partial differential equations yield multiple week solutions and the unmodified discrete equations might not produce correct answers. One way to handle this problem is to spread the shock over some distance over the shock front. The first of this technique was applied by Von Neumann and Richtmyer²⁸ in one dimension problems by adding a viscosity term q in the momentum and energy equations. The multidimensional form of the same concept has the following form:

\begin{align} q & = ρ l (C_{0} l {\dot{ϵ}}_{k k}^{2} - C_{1} a {\dot{ϵ}}_{k k}) i f {\dot{ϵ}}_{k k} < 0 \\ q & = 0 i f {\dot{ϵ}}_{k k} \geq 0, \end{align}

()

where

{\dot{ϵ}}_{k k}

is the trace of the strain rate tensor, l is the characteristic length of the element given as

\sqrt{A}

, in two-dimensional and

\sqrt{V}

in three-dimensional; A is the area and V is the volume of the element, a is the local sound speed, and C₀ and C₁ are dimensionless constants with default values of 1.5 and 0.06, respectively.

4 DAMAGE MODELING

A modified form of the Grady–Kipp damage model²⁹ is used for estimating the damage in brittle solids based on the local tensile strain history and initial flaw distribution given by the 2-parameter (k, m) Weibull distribution. A scalar damage parameter D, (0 ≤ D ≤ 1) is used to scale the stress upon onset of damage. An approximate differential form of D is:

\begin{align} \frac{d D^{1 / 3}}{d t} = \frac{m + 3}{3} α^{1 / 3} ϵ^{m / 3}, \end{align}

()

where

α

is a material constant calculated from three material fracture parameters k, m, and the crack growth speed of the material C_g:

α = \frac{8 π {C_{g}}^{3} k}{(m + 1) (m + 2) (m + 3)} .

()

The effective tensile strain

ϵ

given by Melosh et al.³⁰ is:

ϵ = \frac{σ_{\max}}{K + 4 G / 3},

()

where

σ_{\max}

is the calculated maximum positive principal stress, and K and G are the bulk modulus and shear modulus of the solid, respectively. Accumulation of damage begins when effective strain exceeds a threshold value

ϵ_{\min}

ϵ_{\min} = {(V k)}^{1 / m},

()

where V is the volume of a particle or element. The scaling of the tensile stress is done in the principal stress space where the tensile stress is most easily identified and then transformed back to the Cartesian coordinate space.

5 COUPLING OF FEM WITH SPH

The interaction between FEM and SPH methods begins with generation of imaginary particles in the elements that are near the interaction boundary. If an element is found to be severely distorted then it is replaced by a SPH solid particle so that the simulation can complete with generation of cracks and fragmentation of the solid. The basic theoretical concept in Reference 18 is applied with suitable modifications for fluid–solid interaction problems such as to include high impact blast load and is presented here.

It must be noted that the transfer of forces and maintenance of velocity continuity is not explicitly imposed for SPH calculations but is naturally derived from the corresponding adjacent elements in a manner that is highlighted in the following subsections. Furthermore, any reference to the term “stress” implies the Cauchy stress (the true stress) unless otherwise specified. Any form of “stress” that is used in the formulations described in Sections 3 or 2.2) are changed to the Cauchy Stress by suitable relations and then used where ever calculations with “stress” is required.

5.1 Generation of imaginary particles from Interface elements

For an arbitrary element at the interface boundary as shown in Figure 1(A), the following condition is evaluated:

\begin{align} d_{g i} < η R_{i}, \end{align}

()

where d_gi is the distance from the gravity center of the element to the center of particle i, R_i = 2h_i is the radius of the influence domain of particle i, and

η > 1

is a dilation factor. If there is a SPH particle satisfying this condition, then the element is the potential candidate for the generation of imaginary particle. In this work

η = 2

has been used to generate the imaginary particles.

Details are in the caption following the image — **FIGURE 1**
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The method of determining elements that are treated as imaginary particles and transfer of forces

5.2 Setting properties of imaginary particles

Once the imaginary particles are generated at the element gravity center their properties are to be set so that real SPH particles can interact with them to transfer elemental properties using the kernel approximation as given in Equation (1). The essential properties that are set from the elements are given below:

(a)
The mass and density:
$m_{{IP}_{i}} = m_{E_{I}}, ρ_{{IP}_{i}} = ρ_{E_{I}}$
(b)
The coordinates and velocities:

They are taken as the average of those of the three element nodes.
$x_{{IP}_{i}}^{α} = {\overline{x}}_{E_{I}}^{α} = (x_{ER}^{α} + x_{ES}^{α} + x_{ET}^{α}) / 3$
where R, S, and T are indices of the three nodes of element I
(c)
The radius:
$r_{{IP}_{i}} = \frac{\sqrt{A_{E_{I}}}}{2}$
where $A_{E_{I}}$ is the area of element I.
(d)
The smoothing length:
$h_{{IP}_{i}} = α_{h} A_{E_{I}}$
(e)
The stress:

Is estimated based on the stress at Gauss points.
$σ_{{IP}_{i}}^{α β} = σ_{E_{I}}^{α β}$

the subscripts IP_i indicates ith imaginary particle and E_I indicates Ith element.

5.3 Influence on real SPH particles from adjacent elements near the interaction interface

The forces on a particle near coupling interface are evaluated from their adjacent elements located at the FEM boundary and other real particles. The key idea here is to extend the particle approximations to include both real and imaginary particles in the influence domain of each real particle. In such a case the number of particles N in the summation of Equation (1) would be N_IP + N_RP, where the subscript IP is “Imaginary Particle” and RP denotes “Real Particle.” For example, in Figure 1(A) real particles n₁ − n₁₅ and imaginary particles n₁₆ − n₂₀ are considered as neighbors of particle i. It is noted that the imaginary particles are also used in the calculation of strain rates in Equation (14).

5.4 Forces on element nodes from adjacent particles

In order to maintain stress continuity across the FEM-SPH interface equivalent traction vectors are estimated from the stress of the SPH particles and applied to the attached element edge. A schematic representation with two particles (i and j) is shown in Figure 1(B) the equivalent traction

t_{i, j}^{α}

due to particle i and j, respectively, is expressed as

\begin{align} t_{i, j}^{α} = {σ_{i, j}}^{α β} n^{β}, \end{align}

()

where

n^{β}

is the unit outward normal vector of element side IJ,

σ_{i, j}^{α β}

denotes the stress of the particles i and j, respectively, associated to element side IJ, and N_s is the number of the particles associated to element side IJ. Assuming the points A, B are the projected centers of the particles onto the edge IJ, and considering the shape function H matrix as;

H_{I J} = [\begin{matrix} H_{I}^{A} H_{J}^{A} \\ H_{I}^{B} H_{J}^{B} \end{matrix}],

()

where

H_{i}^{A}

implies the value of the shape function of node I at point A, the traction at nodes I and J is given as:

[\begin{matrix} t_{I} \\ t_{J} \end{matrix}] = H_{I J}^{T} [\begin{matrix} t_{i} \\ t_{j} \end{matrix}]

()

this traction to act uniformly and perpendicularly onto the side IJ with nodal values of this traction being, t_I = t_J = t_IJ, the nodal forces on element side is:

\begin{align} [\begin{matrix} F_{I}^{α} \\ F_{J}^{α} \end{matrix}] = \int_{I J} H_{I J}^{T} [\begin{matrix} t_{I} \\ t_{J} \end{matrix}] d l_{I J} \end{align}

()

which is the same procedure for standard FEM calculations.

5.5 Association of SPH particle with element edge

A particle near an element edge must be identified for transferring stress, velocity and other data from elements to particles and vice-versa. Figure 1(C) shows a schematic diagram of how the particle is determined for an element edge. In this figure particle i is associated to element side IJ if the following criterion is satisfied:

\begin{align} ξ_{l} \leq R_{N} \leq ξ_{h}, 0 \leq R_{T} \leq 1, \end{align}

()

where

R_{N} = δ / r_{i}

δ

is the penetration distance between particle i and element side IJ,

r_{i} = 0.5 h_{i} / α_{h}

is the radius of particle i and R_T = l_JO/l_JI ,

l_{J O} = x_{O J}^{α} x_{I J}^{α} / \sqrt{x_{I J}^{α} x_{I J}^{α}}

l_{J I} = x_{I J}^{α} x_{I J}^{α} / \sqrt{x_{I J}^{α} x_{I J}^{α}}

x_{O J}^{α} = x_{O}^{α} - x_{J}^{α}

x_{I J}^{α} = x_{I}^{α} - x_{J}^{α}

, and

ξ_{l}

and

ξ_{h}

are two specified constants. In Figure 1(C) point O is the projection of particle i on element side IJ, and the penetration distance

δ

is defined as

\begin{align} δ = (a x_{i}^{1} + b x_{i}^{2} + c) - r_{i}, \end{align}

()

where

a = x_{J}^{2} - x_{I}^{2}, b = x_{I}^{1} - x_{J}^{1}

, and

c = x_{I}^{2} x_{J}^{1} - x_{I}^{1} x_{J}^{2}

δ

is negative when particle i crosses over element side IJ and positive when gap exists between particle i and the element side.

The criterion described above allows some penetration and gap between the element side and its associated particles. The allowed penetration and gap are controlled by constants $ξ_{l}$ and $ξ_{h}$ . In this work, $ξ_{l}$ and $ξ_{h}$ are taken as −1 and 1, respectively. This makes the penetration and gap less than the particle radius.

The association between particles and element sides is determined at the beginning of computation. It might change during the computational process. At certain intervals the association between the particles and element edges needs to be redetermined. Since the particles that have been once associated with an element side might get detached from the edge and new particles may take up their place. Hence element to particle association near the coupling interface has to be reevaluated at certain intervals of the computational cycle.

5.6 Adaptive coupling technique

The initial computational model, with the exception of the explosive domain, is entirely composed of elements. During the computation, severely distorted elements are converted to particles and the generated particles near coupling interfaces are linked to the elements by the coupling algorithm.

The adaptive coupling includes four steps:

Decide on the level of distortion of an element for which it should be converted to an particle.
Establish afresh the list of elements that are on/near the coupling interface and that are to be treated as imaginary particles.
Replace distorted elements with their real particle counterparts, and generate the new set of imaginary particles.
Update the coupling surface of element regions and establish new association between particles and element sides.

This procedure is listed as Algorithm 1.

5.6.1 Determination of elements to be converted to particles

5.6.1.1 Geometrical quality

For triangular element the quality $ν$ of a triangle is checked as two times the ratio of the radius of the inscribed circle divided by that of the circumscribed circle (an equilateral triangle achieves the maximum possible quality of 1). If $ν$ is less than a critical value $ν_{c} = 0.6$ , the element is considered to be distorted and recorded as one converted to a particle.

5.6.1.2 Material quality

An element with damage parameter D = 1.0 is completely incapable of sustaining tensile loads. Hence if for an element D ≥ 0.9 then this element is also recorded as the one to be converted to particle.

A group-based conversion (GBC) scheme is combined with the quality criterion to determine elements converted to particles. In the GBC scheme, elements of each body are divided into groups at the beginning of the computation. During the computational process, if one or more elements in a group satisfy the conversion criterion, then all elements in that group are converted to particles.

In order to classify the elements into groups a minimum rectangular domain is first obtained to envelop the body. The rectangular domain is then divided into uniform subdomains with reference size

Δ_{s}

, which is determined as

\begin{align} Δ_{s} = η R_{\max}, \end{align}

()

where

η

is a dilation factor which is the same as that in Equation (48), and R_max denotes the maximum possible radius of influence domains of particles in case all elements are converted to particles. R_max is estimated by

\begin{align} R_{\max} = 2 h_{\max} = 2 α_{h} \sqrt{A_{0 \max}}, \end{align}

()

where A_0max is the maximum initial area of all elements. After determining

Δ_{s}

, the number of subdomains in

x^{α}

-direction,

N_{x^{α}}

, can be calculated by

\begin{align} N_{x^{α}} = \max (I n t [(x_{\max}^{α} - x_{\min}^{α}) / Δ_{s}], 1) \end{align}

()

and the exact sizes of subdomains can be given as

\begin{align} Δ x^{α} = ((x_{\max}^{α} - x_{\min}^{α}) / N_{x^{α}} . \end{align}

()

The subdomains are numbered as

(I_{x^{1}}, I_{x^{2}})

, where

I_{x^{1}}

ranging from 1 to

N_{x^{1}}

is the index in x¹-direction and

I_{x^{2}}

is similar, see Figure 2. An element is considered to be in a subdomain when its gravity center is in the subdomain. The number of the subdomain where an element is located can be easily determined by:

\begin{align} I_{x^{α}} = I n t [(x_{g}^{α} - x_{\min}^{α}) / Δ x^{α}] + 1, \end{align}

()

where

x_{g}^{α}

are coordinates of the element gravity center. With Equation (59), each element is sorted into a unique subdomain. Then, all the elements in the same subdomain compose an element group.

5.6.2 Determination of elements treated as imaginary particles in adaptive coupling

In the adaptive coupling process, the positions of coupling interfaces will change when some elements are converted into particles. Thus, elements near coupling interfaces, which are required to treat as imaginary particles for coupling calculation, will also change and should be redetermined. Assuming that element A in group (I, J) satisfies the conversion criterion with

ν_{c}

, all elements in group (I, J) are converted to particles, and all elements in its neighboring groups are treated as imaginary particles. According to the numbering rule in Equation (59), the neighboring groups of group (I, J) are easily obtained as

\begin{align} (I \pm 1, J), (I - 1, J \pm 1), (I + 1, J \pm 1), (I, J \pm 1) . \end{align}

()

There are generally eight neighboring groups for a given group unless it is a boundary group. Among these neighboring groups, some also should be or have been converted particles. In these cases, they are excluded from the groups treated as imaginary particles.

5.6.3 Update of surfaces of element regions and association between particles and element sides

When subdomains are eroded the FEM-SPH interfaces are updated. At the end of each computational cycle the association between interface particles and interface element edges is checked. If a subdomain is eroded and replaced by equivalent real SPH particle then a new list of particles and element edge is created by the same procedure as described in Section 5.5 above. Similarly the list of imaginary particle created from elements is also updated (cf. Section 5.1). After updating interfaces and generating new real particles from elements the next computational cycle continues.

6 NUMERICAL TESTS AND RESULTS

The above procedures are implemented in a computer code written by the authors using Object Oriented Programming Techniques with C++. In this article, the proposed coupling procedure is verified by analyzing elastic impact on a block and the results are compared with compared with analytic/published solution for verification of the procedure. This article also describes the simulation of an explosion in brittle rock and elaborates the coupling mechanism between product gas particles and solid elements. The details are given in the following sections.

6.1 Elastic impact

For validation of the above coupling algorithm an example calculation is presented here. The example is taken from Reference 24. A block of material is assumed under plane-strain conditions Figure 3 of dimensions L = 5 mm and W = 10 mm. The Young's modulus E = 3.24 GPa, the Poisson's ratio $ν = 0.35$ , and the mass density $ρ = 1190 {kg m}^{- 3}$ that gives a dilatation wave speed of $c_{d} = 2090 {m s}^{- 1}$ . The block is not constrained and is loaded by an impact velocity which acts in the positive y-direction on the top surface of the block, at y = +L. The impact velocity is increased linearly from v_p = 0 to 10 m s⁻¹ in time t_r = 1.0 × 10⁻⁷s. Because of symmetry with respect to the y-axis only one half of the block has been modeled. The mesh consists of 20 × 40 quadrilateral elements for the model simulated only by FEM. For the coupled calculation 800 triangular elements and 2165 particles were used (that includes 1600 real particles and 565 background imaginary particles are discretized as shown in Figure 4(C). The displacements of the nodes on the symmetry axis are constrained in the x-direction. The total time of analysis was 3.09 $μ s$ and time step $Δ t$ is set to 1.0 × 10⁻⁸ which satisfies the CFL criterion. The stress contours along the y-axis at different times are given in Figure 4(B), the FEM model and Figure 4(C), the coupled FEM-SPH model. As observed that the plots are almost identical to the one given in Reference 24, Figure 3. The oscillations at the wave front which is due to initiation effects appear before the shock in Figure 4(A) and after the shock in Figure 4(D,E). This can is attributed to the fact that HHT $- α$ parameters damps out the numerical noise from the nonzero solution more effectively as compared with the central difference method used in Reference 24, Figure 4(A), which has on such dissipative mechanism. Hence this procedure can be used for analyzing high impact problems such as rock blasting, where otherwise handling mesh distortion becomes critical.

The analytic solution for the magnitude of the stress carried by the pulse is given by $σ_{y y} = ρ c_{d} Δ v_{p}$ which turns out to be equal to 24.81 ≈ 25 MPa. This value is also obtained from our simulations, Figure 4(D), FEM and 4(E), FEM-SPH coupled models. Furthermore, the speed of the stress wave equals the distance traveled by the first peak divided by the elapsed time. From Figure 4(D,E) it is seen that the first peek traveled almost 6.5 mm in $3 μ s$ from which the calculated speed of the pulse is ≈2166 m s⁻¹, which is close to the dilatation wave speed $c_{d} = 2090 {m s}^{- 1}$ . All these results indicate that the interaction between the SPH particles and the finite elements in the coupled FEM-SPH procedures works well in simulating an impact phenomenon.

6.2 Blast in square hole

In this example, a simple example of explosive blast using TNT filled in a square hole in a square brittle rock is modeled under plane strain conditions. Here, explosive in the blast hole is discretized with SPH particles and rock part is initially modeled with finite elements as shown in Figure 5(A).

For the product gas the stress component in Equations (2) and (3) is defined as:

\begin{align} σ^{α β} = - p δ^{α β} \forall x \in Ω_{t}^{g a s} . \end{align}

()

Equation (61) requires the equation of states (EOS) for pressure of gases for completeness of the system of equations in Equation (1). The EOS of the explosive after detonation is assumed based on “Jones–Wilkins–Lee” as given by:

p = A \cdot (1 - \frac{ω}{R_{1} \cdot W}) \cdot \exp (- R_{1} \cdot W) + B \cdot (1 - \frac{ω}{R_{2} \cdot W}) \cdot \exp (- R_{2} \cdot W) + \frac{ω \cdot e}{W},

()

where ratio

W = ρ_{e} / ρ

is defined by using

ρ_{e} =

density of the explosive (solid part) and

ρ

= density of the detonation products. The parameters A, B, R₁, R₂ and

ω

are given by several References 5, 31, 32. In addition, the initial density of solid explosive

ρ_{0}

, speed of detonation V_D, Chapman–Jouguet pressure P_CJ and the chemical energy of the explosive per unit mass e with initial value e₀ are given in Table 1, from Reference 31.

TABLE 1. The Jones–Wilkins–Lee parameters for some high explosives

Material	$ρ_{0}$ g cm⁻³	V_D m s⁻¹	p_CJ (GPa)	A (GPa)	B (GPa)	R₁	R₂	$ω$	e₀ J kg⁻¹
TNT	1.630	6930	21.0	373.8	3.747	4.15	0.90	0.35	6 ×10⁶
Composition B	1.717	7980	29.5	524.2	7.678	4.20	1.10	0.35	8.5 ×10⁶
PBX 9501[9]	1.844		36.3	852.4	18.02	4.55	1.3	0.38	10.2 ×10⁶

After detonation, solid explosive particles converts into gas and exert pressure to solid boundary based on Equation (61) The external dimension on the model are 0.1m × 0.1m with a centrally place square hole of size 0.01m × 0.01m of the model. The material properties used were, density = 2500 kg m⁻³, Young's modulus E = 25 GPa, and Poisson's ratio R = 0.1. The Weibull parameters for damage evolution are taken to be, k = 1.59 × 10³⁸, m = 9.5, $C_{g} = 600.0 {m s}^{- 1}$ corresponding to that of a brittle rock. The hole is filled with TNT of properties given in Table 1.

Initially, the FEM model consists of 39,600 constant-strain triangular solid elements and 576 explosive particles with an arrangement as shown in Figure 5(A). The simulation begins with detonation of explosive particles and the pressure generated by it is applied to the FEM-SPH interface based on the developed coupling algorithm. Then the kinematic variables such as displacements, velocity and accelerations as well as stress are obtained in the finite elements by the method given in Section 5 and at the same time these variables are also updated in the imaginary particles. The variables of the real SPH particles are updated considering all near neighbor real and imaginary particles. The detailed steps of this procedure is given in Algorithm 1.

Figure 5(B) shows the initial discretization of the model with elements and particles and Figure 6(A-G) show the subsequent evolution of the coupling interface in terms of the X-velocity $(v_{x})$ and damage (D) after blast depicting element replacement with time.

The velocity and damage parameters are further analyzed for three selected elements at three different locations in the model to highlight the efficacy of the developed procedure in replacement of elements by real particles. Figure 7(A) shows the location of these three elements, one at the rock-gas interaction interface (element number 19617), one at the free face (element 19443) and one at the middle of the rock mass. The average nodal velocity profile of these elements and their corresponding velocity profile as particles after damage is shown in Figure 7(B).

It is observed that the element number 19617, that is closest to the explosive, is damaged immediately after the blast and is eroded from the FEM model. For subsequent calculations it is treated as real SPH particle. Element number 19545 is converted to a real SPH particle at about $1.2 μ s$ but is completely damaged at $4.83 μ s$ . This is because its erosion from FEM model had occurred due to some other element in its group reaching the threshold value of the erosion criteria at time $1.2 μ s$ . The same phenomenon is observed for element number 19443. It is also noted that although element number 19545 is closer to the explosive than the element number 19443 the latter is damaged at a much later stage than the former. This happens because element 19443 is located at the free boundary of the rock mass and thus more susceptible to tensile failure. Figure 7(B) also demonstrates that after damage as the elements are replaced with real SPH particles velocity reached to a more or less constant value.

The behavior of gas particles in terms of pressure and X-velocity variations is investigated by considering three particles (cf. Figure 8(A)) located in the middle of the blast hole (particle 276), at the rock-gas interaction interface (particle 12) and at an intermediate point (particle 108) as depicted in Figure 8(B,C). It is observed that all the particles begin with an initial high pressure of about 8.37 × 10⁹ Pa which gradually decays to a residual value with time. A closer look of the pressure data reveals that within a few microseconds after the blast both the pressure and the velocity change rapidly in all the particles (cf. Figure 8(B,C)). This may have happened because of the fact that the initial velocity of the product gas is much larger than that of the surrounding rock mass. As a result, the outward movement of the expanding gas particles is obstructed by the rock causing the particles to return back towards the center of the blast hole. Pressure in the gas particles drops as the velocity of the particles changes its direction. After damage, the rock particles moves outwards keeping up with the velocity of the gas particles. The expansion of the gas causes reduction in the velocity. After some time the variations in velocity and pressure becomes almost negligible. This physical phenomenon is aptly captured by the simulation process.

7 CONCLUSION

This work attempts to alleviate some of the shortcomings of the standard Lagrangian-FEM by introducing particles of the SPH method, on the fly, in regions where mesh entanglement occur. This adaptive coupling algorithm between SPH and FEM is proposed in this article for analyzing high impact shock inducing problems. This is done in a way such that traction and velocity boundary conditions are maintained at the interface boundary, using imaginary particles that complete the kernel approximation for SPH calculations at these boundaries. In addition, a continuum damage model predicts the onset and distribution of failure in solids. As a result, mass and momentum of the total system is conserved. The article also describes the detailed implementation of this adaptive coupling procedure providing an example of elastic bending of beam for validation. Furthermore, an example from blasting of brittle rocks, where fluid–solid interaction is involved under high impact blast load is simulated. The proposed coupling algorithm can be applied to large-scale problems where such fluid–solid interaction is perceived considering geometric and material nonlinearity. However, since there are several process involved (cf. Algorithm 1) when erosion of elements take place this procedure is time consuming and hence effort must be given to develop an efficient algorithm for attaching particles with element edges.

ACKNOWLEDGMENTS

We acknowledge Department of Science and Technology (DST), India and Coal India Limited (CIL) for the partial financial support to this research.

CONFLICT OF INTEREST

The authors declare no potential conflict of interests.

APPENDIX

Algorithm 1. Adaptive coupling algorithm

Biographies

Arijit Khan Graduated as Civil Engineer from Jadavpur University Kolkata, India, his interest in Computational Mechanics impelled him to enroll in the PhD program at Indian Institute of Technology, Kharagpur, India, Department of Mining Engineering under the stewardship of Prof. Debasis Deb with very similar passion for the subject. He completed PhD in the year 2019 and is currently involved in simulation software development for use in oil and gas industry.
Dr. Debasis Deb received his PhD degree from the University of Alabama, Tuscaloosa, USA in 1997. Currently, he is serving as a Professor in Department of Mining Engineering, IIT Kharagpur. He has work experience as Research Associate at Korea Institute of Geoscience and Mineral Resources (KIGAM), South Korea for 1 year. Dr. Deb is the recipient of prestigious National Geoscience Award (2013), presented by The President of India, for his works in the area of rock mechanics and ground control. He has published two books and a chapter of a book. He has published over 180 research articles in various national and international journals and conferences and has five patents in his credits.

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Citing Literature

All articles

An adaptive coupling of finite elements with smooth particle hydrodynamics particles for large deformation fluid–solid interactions

Abstract

1 INTRODUCTION

2 THE SPH METHOD

2.1 The governing equations

2.2 Constitutive relation

2.3 Time integration in particle method

2.4 Determination of time step size

3 LAGRANGIAN FORMULATION FOR LARGE DEFORMATION ANALYSIS

3.1 The generalized– α method

3.2 Shock resolution by artificial viscosity

4 DAMAGE MODELING

5 COUPLING OF FEM WITH SPH

5.1 Generation of imaginary particles from Interface elements

5.2 Setting properties of imaginary particles

5.3 Influence on real SPH particles from adjacent elements near the interaction interface

5.4 Forces on element nodes from adjacent particles

5.5 Association of SPH particle with element edge

5.6 Adaptive coupling technique

5.6.1 Determination of elements to be converted to particles

5.6.1.1 Geometrical quality

5.6.1.2 Material quality

5.6.2 Determination of elements treated as imaginary particles in adaptive coupling

5.6.3 Update of surfaces of element regions and association between particles and element sides

6 NUMERICAL TESTS AND RESULTS

6.1 Elastic impact

6.2 Blast in square hole

7 CONCLUSION

ACKNOWLEDGMENTS

CONFLICT OF INTEREST

APPENDIX

Algorithm 1. Adaptive coupling algorithm

Biographies

REFERENCES

Citing Literature

Figures

References

Related

Information

3.1 The generalized– $α$ method