EDA

Blast simulation using advanced Finite Element Methods

2012-12-31T03:11:00.000-05:00

We earlier discussed the differences between Implicit and Explicit analysis and Newton Raphson method for implicit analysis. In this blog post we will discuss advanced simulation methods for performing a blast loading simulation on a simple steel plate. There are numerous application areas for performing a blast simulation, some examples are (a) designing military/infantry vehicles capable of resisting mine blast explosions, (b) design of blast resistant concrete structures or (c) design of bridges. It is not feasible to test the designs directly by conducting a physical/field blast test unless the design is fool proof, it would result into tremendous costs if primitive designs are tested on field. A finite element simulation is thus a cost effective option for designing blast resistant structures.

Lets consider an example of vehicle subjected to a mine blast for the purpose of understanding. In this simulation, a TNT explosive charge would be placed on ground beneath the vehicle. For the purpose of simplicity we would consider a simple plate and a small charge of 1 kg TNT. There are two most common methods to simulate the blast loading effect:

Use of a simplified CONWEP function
Use of Arbitrary Lagrangian Eulerian (ALE) method

Article in progress. Meanwhile please refer my thesis on blast simulation: http://bit.ly/thesis_amogh

Arbitrary Lagrangian Eulerian (ALE) method in LS Dyna

2012-11-11T23:47:00.001-05:00

The finite element method which we generally use is termed as a lagrangian formulation. Although the normal methods tends to be feasible for usage in short duration impact applications such as crash analysis, applications concerned with very short transient loadings, such as blast analysis require better simulation algorithms. The use of plain Lagrangian Formulation in large deformation problems leads to severe distortion of mesh rendering it incapable of yielding accurate results.

Also the time step size per iteration becomes very small and consequently increases the computational time. To some extent high deformation problems can be modeled by Lagrangian techniques by using 1) Adaptive re-meshing algorithm or 2) Element deletion method. But the adaptive re-mesh algorithm prohibitively increases the computational time and is also not fully developed for three dimensional problems. The deletion of elements results in significant loss of mass. Hence there is a need for an element formulation which would avoid the mesh distortion and also be computationally efficient. The Arbitrary Lagrangian Eulerian formulation also referred to as ALE formulation is used in large strain deformation problems, mainly because it does not face problems regarding mesh distortion. This method is a combination of Lagrangian as well as Eulerian methods.

In ALE formulation, an arbitrary referential domain is defined for the description of motion which is different from the material (lagrangian) and the spatial (eulerian) domain. In the lagrangian method the mesh moves along with the material where as in the Eulerian method the mesh remains fixed and the material moves through it. Also more than one material can be included in an Eulerian formulation. In cases where there are multiple Eulerian materials, Multi Material Groups (known as ALE-MMG) are required to be used. Each of the Eulerian material is assigned to a group using the LS Dyna keyword *ALE_MULTI_MATERIAL_GROUP. It is to be noted that each part within a specific multi material group has identical material properties.

The ALE formulation can be defined as an algorithm for performing an automatic rezoning. Manual rezone consists of the following three steps:

Stop Calculations after Mesh Distortion
Mesh Smoothing
Advection - Remap the solution from Distorted Mesh to Smoothened Mesh.

1. Stop Calculations after Mesh Distortion: The calculations are stopped after the distortion of mesh so as to perform the mesh smoothing.

2. Mesh Smoothing: There are many algorithms available for mesh smoothing. Some of them used in LS Dyna are listed below:

Simple Averaging: In this algorithm the coordinates of a node is the simple average of the its surrounding node coordinates.
Kikuchi’s Algorithm (Volume Weighted Averaging): This algorithm uses a volume-weighted average of the coordinates of the centroids of the elements surrounding a node.
Equipotential Smoothing: Equipotential zoning is a method of making a structured mesh for finite difference or finite element calculations by using the solutions of Laplace equations as the mesh lines. This same method can also be used to smooth selected points in an unstructured three-dimensional mesh if it is at least locally structured.
Surface Smoothing: The surfaces are smoothed by extending the two-dimensional equipotential stencils to three dimensions.
Combining Smoothing Algorithms: In LS Dyna, the user can use a weighted average of the three algorithms (Equipotential, Simple averaging and Kikuchi‟s smoothing algorithm) to generate a composite algorithm.

3. Advection: The remap or the advection step maps the solution from a distorted Lagrangian mesh on to the new mesh. During the remap, the topology of the mesh is assumed to be fixed. Also the mesh motion during a step is assumed to be less than the characteristic lengths of the surrounding elements. In other words the Courant number should be less than or equal to one.

Where, C is the Courant‟s Number, u is Velocity, t is Timestep and x is the Length Interval. Generally during the advection the following steps are performed:

a) Decide which nodes to move.

b) Move the boundary nodes.

c) Move the interior nodes.

d) Calculate the transport of the element-centered variables.

e) Calculate the momentum transport and update the velocity.

Each element solution variable needs to be transported. The total number of solution variables is dependent on the material models. A minimum of six variables need to be transported. The number of variables may go up to fifteen when the elements have strength, where components of the stress tensor and the plastic strain etc. must also be advected. The following schemes can be used for the purpose of advection:

Van Leer advection scheme
Donor Cell advection scheme

The Van Leer advection scheme is a Second order monotonic scheme whereas the Donor Cell is First order accurate scheme. During the transport calculations the errors may bring about the smoothing of the solution, reducing the peak solution variable values. The monotonic second order advection algorithms prevent the transport calculations from creating new spurious oscillations or the peak values (either minimum or maximum) for the variables. Hence Second order accuracy is required for transport of variables. The monotonic algorithms generally provide stability to the code. Also there are many constitutive material models which have a specific allowable range for their history variables. The constitutive models are defined only for the specific range. The second order advection scheme is generally used for supersonic Eulerian flows, where the first order advection schemes may be inaccurate.

Note: This article is still in progress. Sorry for the complicated wordings.

Reference: LSTC Theory Manual, J. Hallquist

Newton-Raphson method for Implicit Analysis

2012-05-16T23:21:00.002-04:00

Few months back I blogged about difference between explicit dynamic and implicit analysis. Concisely summarizing, an implicit analysis is the one which is independent of time and explicit is the one which is of a short duration and time dependent. In this post, I would be going into details regarding an Implicit Finite Element Analysis. Newton-Raphson (NR) method is the commonly used method for most of the implicit FE solvers. In simple numerical terms, the N-R method is used for finding successively better approximations. Figure 1 shows the general iterative process for an implicit analysis. Click here to read more about N-R method.

Figure 1: General Iterative process for Implicit Analysis

The Regular and the Modified method are two variants of Newton-Raphson method. Both of them iteratively increment the displacement vector. The equation (1) is used by both the methods to determine the increments. Where K_i is the stiffness matrix which represents the tangential stiffness of the structure, 'u' is the displacement and 'g' is the out of balance force. The evaluation of stiffness relation in equation (1) forms a basis for differentiation between Regular and Modified NR method.

K_i =

equation (1)

Regular Newton-Raphson method: In the Regular Newton-Raphson iteration, the stiffness relation (equation 1) is evaluated every iteration as shown in Figure 2. This means that the prediction of change in displacements is based on the last known or predicted situation which necessarily need not be an equilibrium state.

Figure: Regular Newton-Raphson iteration (Ref: TNO Diana user manual)

This method brings convergence within only a few iterations. However the main disadvantages of the method include time consumption and accuracy of the solution. As the stiffness matrix has to be setup for every iteration, using a direct solver can result into heavy computational costs as a consequence of decomposition of the matrix. The accuracy of the solution would depend on the initial prediction or guess, if it is far from the final solution, the method easily fails because of divergence. Thus, the regular method requires fewer iterations but is relatively time consuming.

Modified Newton-Raphson method: The Modified Newton-Raphson method only evaluates the stiffness relation (equation 1) at the start of the increment as shown in figure 3. The prediction is always based on a converged equilibrium state as opposed to that of the regular method. This method converges slower than the regular method, however, for every iteration setting up of a new stiffness matrix is not required. Generally speaking, the modified method would require larger number of iterations but each of the iterations take less time as compared to the regular method.

**Figure 3:** Modified Newton-Raphson iteration (Ref: TNO Diana user manual)

The Modified Newton-Raphson process can sometimes converge in situations where the regular method would not. For more reading, kindly refer to the links below:

Kindly ignore the formatting, blogger editor gave me a hard time. Time to switch to wordpress.

Note: Most of the material has been referred from the TNO Diana manual. I do not claim any copyrights. If you are interested in improving the content, kindly let me know, I can grant you access to the post.

MADYMO - Occupant Safety Simulation Software

2012-03-04T21:20:00.000-05:00

MaDyMo stands for Mathematical Dynamic Models and is an Occupant Safety Simulation software. It is created by TNO Automotive Safety Solutions (TASS). Madymo is equipped with a multibody (MB) dynamics, finite element (FE) as well as a computational fluid dynamics (CFD) solver. The primary application of Madymo is to study the safety of occupants and injury risks in a vehicle crash scenario.
However, the applications can be extended to aviation, defense, accident reconstruction simulations for example. You can also perform a crash analysis using madymo. While other softwares offer the capability to perform occupant safety simulations, Madymo can simulate the same problem efficiently consuming only one fourth of the computational time. Madymo uses a unique combination of rigid body dynamics and finite elements making the simulations run faster. The FE solver is based on explicit dynamic numerical techniques.

Madymo can also be coupled with other crash solvers like LS-Dyna, Pam-crash, Abaqus and Radioss in order to utilize the FE capabilities of the third party solvers along with the multibody and airbag modeling techniques of Madymo. Some of the madymo case studies and tips/tricks can be found here.

True, Engineering and Effective Stress Strain

2011-12-19T14:57:00.002-05:00

True Stress - Strain Curve:
The engineering stress-strain curve is based entirely on the original dimensions of the specimen hence does not give a true indication of the deformation characteristics of a metal because. The dimensions change continuously during the test..

Read more: http://bit.ly/uxysCM

Converting True Total Strain to Effective Plastic Strain:
Some material models require Effective Plastic strain as their input values. Go to the link below for more details.

Read more:http://bit.ly/tdN7bo

Article still in wip stage

Cyber infrastructure in engineering design

2011-06-28T13:53:00.001-04:00

Multiobjective Optimization and Trade Offs Using Pareto Optimality

2011-06-28T13:47:00.004-04:00

LS Prepost 3.2 released for mac

2011-06-16T12:19:00.004-04:00

LS-PrePost an advanced pre/post-processor delivered free with LS-DYNA has now a Mac version. LS Prepost supports all the keywords for LS Dyna. Other pre processors that can be used for creating an input deck for LS Dyna include Hypermesh, Oasys, Ansys etc. LS-PrePost runs on Windows, Linux, and Unix utilizing OpenGL graphics to achieve fast rendering and XY plotting. The latest build can be downloaded from LSTC's FTP Site. The tutorials and online documentation for LS Prepost can be found here.
The online documentation website has a wide range of tutorials for applications such as impact, dummy positioning, Metal forming etc.

Please click here to download the Mac version.

Image Courtesy: D3View blog, Suri Bala

Consistent Unit System in LS Dyna

2011-05-20T09:13:00.010-04:00

Explicit Dynamics software LS Dyna has no default unit system. The user needs to provide all the values in a consistent system according to which the solver gives results. If 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

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

Implicit and Explicit Finite Element Analysis

2011-04-03T11:07:00.008-04:00

Implicit and Explicit are two types of approaches 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 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 Hyperelastic Finite Elements

2011-03-24T17:04:00.008-04:00

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 of 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

2011-03-18T10:52:00.001-04:00

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

2010-05-30T19:37:00.001-04:00

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