>Consistent Unit System in LS Dyna

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Explicit Dynamics software LS Dyna has no default unit system. The user need to provide all the values in a consistent system according to which the solver gives results. It the user inputs mass in Tonnes and length in meters, the solver will assume the length to be in ‘mm’

The Definition of a consistent system of units as explained by Dynasupport is as follows:

• 1 force unit = 1 mass unit * 1 acceleration unit
• 1 acceleration unit = 1 length unit / (1 time unit)^2
• 1 density unit = 1 mass unit / (1 length unit)^3

The following table displays the units which be used together:

Consistent Unit Systems in LS Dyna

[http://www.dynasupport.com/howtos/general/consistent-units]

>Implicit & Explicit Finite Element Analysis

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Implicit and Explicit are two types of approached that can be used to solve the finite element problem. The implicit approach is useful in problems in which time dependency of the solution is not an important factor [e.g. static structural, harmonic, modal analysis etc.] whereas Explicit Dynamics approach is most helpful in solving high deformation time dependent problems such as Crash, Blast, Impact etc.
      

The prime difference between the implicit and explicit scheme lies in the consideration of velocity or acceleration. You must be aware of the equation relating mass (m), damping (c), stiffness (k) and force (F). In equation 1, ‘x’ stands for displacement whereas ẋ and ẍ are resp. the first and second time derivatives of ‘x’. In other words they stand for velocity and acceleration resp.

mẍ + cẋ + kx = F                                (1)

Implicit Scheme: In an implicit scheme, the displacement is not a function of time (i.e. x = constant). Hence the velocities and accelerations which are time derivatives of displacement turn out to be zero and the mass and damping factors can be neglected. The implicit method can be based based on Newark’s method, Newton Raphson Method etc. In order to solve an FEM problem using implicit method, inversion of stiffness matrix (k) is required. Very Large deformation problems such as crash analysis can result in millions of degrees of freedom effectively increasing the size of stiffness matrix. Larger the stiffness matrix longer is the computational time required for its inversion. Hence there is a need for an explicit method which would prevent the inversion of stiffness matrix. Implicit methods are mainly used in softwares such as Ansys, Nastran, Abaqus etc. 

Explicit Scheme: As opposed to Implicit methods, explicit scheme is a function of time. Being a function of time, the velocity and acceleration as well as the mass and damping need to be considered in this scheme. In an explicit method, Central Difference time integration (CDTI) is used to calculate field variables at respective nodal points. Since only a numerical solution is possible for a non linear ordinary differential equation, this method is particularly suited for non linear problems. It requires the inversion of the lumped mass matrix as opposed to that of the global stiffness matrix in the implicit methods. In the CDTI, the equation of motion is evaluated at the previous time step (t_n-1, where t_n is the current timestep). 

The explicit method or algorithm works in timestep increments i.e. the displacements are calculated as the time proceeds. Consider the simulation of a crash analysis. At timestep 1 (t=0 ms), there is no deformation since the impact is yet to occur. Gradually as time would progresses the deformation also would change. Assume that at timestep 2, t is 5 ms, now the explicit algorithm will calculate the values of field variables at time when t=5 ms. This is the way in which the solution proceeds.

LS Dyna is one software which is based on explicit dynamics and is especially used for solving problems such as Crash, Impact, Penetration etc. Pam crash and Abaqus explicit are also based on the same.

 

>Volumetric Locking in Finite Elements

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Volumetric Locking is exhibited by incompressible materials such as rubber having poisson’s ratio near or equal to 0.5 resulting in an overly stiff response. The overly stiff response is depicted in figure below.

The mathematical reason can be explained as below: 

The stress can be divided into Deviatoric (distortional) and Volumetric (dilatational) components. The volumetric component is a function of bulk modulus and volumetric strain. If you have a look at the basic equations for bulk modulus, it can be observed that with poisson’s ratio as 0.5 the bulk modulus shoots to infinity resulting in overly stiff behavior. It has been observed that Fully Integrated Elements are prone to Volumetric Locking.

Stress = -pI + s; 

where p=> Volumetric Component [Hydrostatic pressure]

          s=> Deviatoric Component

p = -K * (Volumetric Strain);

K = E/ [3(1-2v)];

where K=> Bulk modulus

          v=> Poisson’s ratio

Causes for Higher Tendency of Volumetric Locking in Fully Integrated Elements:

– At each integration point of the element the volume remains almost the constant and hence overconstraints the kinematically admissible displacement field.

– Consider an 8 noded 3D Hexahedral element. A Fully integrated element would have 8 integration points per element resulting in 8 constrains/element. But in this case only 3 d.o.f are available to satisfy these constraints.

– This results in overconstraining of mesh, also known as “Locking“

How to Avoid Volumetric Locking:

1] Use Reduced Integration: It has fewer volumetric constrains

2] Use of Selective Reduced Integration: It treats the volumetric and deviatoric parts pf stiffness matrix separately.

3] Use of B-Bar method: Similar to selective reduced integration. But instead of separating volume integral into two parts, the definition of strain is modified.

4] Use of Hybrid Elements: They work by including the hydrostatic stress distribution as an additional unknown variable, which must be computed at the same time as the displacement field.  This allows the stiff terms to be removed from the system of finite element equations.

5] Reduced Integration with Hourglass Control: Artificial stiffness is added to the element which constrains the hourglass mode.

For more details, the reader is referred to Solid mechanics by  A.F. Bower [http://solidmechanics.org/text/Chapter8_6/Chapter8_6.htm#Sect8_6_2]

>Non Linear Hyperelastic Finite Element Analysis

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This post would share Need To Knows & discuss some tips and tricks related to Non Linear HyperElastic FE Analysis.

NeedToKnow#1: The poisson’s ratio of Rubber although being 0.5 is considered as 0.4955 in a Finite Element Analysis. The reason being that the Bulk modulus becomes infinity at Poissons ratio of 0.5. Also the volumetric strain equals near zero.

NeedToKnow#2: Nearly or Fully Incompressible Material (with Poisson’s ratio nearly 0.5) exhibit Volumetric Locking in Fully Integrated Elements

>Engineering Design & Analysis

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This Blog is intended to share the knowledge in the Engineering Design and Finite Element Analysis domain.