The typical Finite Element Method has been used with success in the industry as well as research for many years. However, there are certain situations where the mesh-based approach is not optimal. For example, simulations where the model undertakes large deformations are hard to deal with a mesh-based approach as the mesh becomes severely distorted leading to high inaccuracies and most likely termination of the simulation. One strategy to overcome this problem is remeshing, which means that the mesh is to be recreated during the simulation. However this approach is costly, often difficult to implement in 3D, and prone to errors as the simulation progresses.

Overcoming of the typical FE discretization disadvantages naturally led to the idea of mesh free methods where the domain is no longer to discretized using elements, but instead a set of data points – “nodes” – are used. Meshless methods are useful for a diversity of applications, but usually come with an increase in computational time required to run the simulations.

Liu [1] addressed the **needs for mesh free methods**, which **include the** **handling of large deformations problems**, the **avoiding of stability issues**, the **representation of** **continuous variable fields** **and fluid representation and its interaction with structures**. Some of these methods are the Element Free Galerkin (EFG), Finite Point Method, Point interpolation method, Boundary node method, SPH, etcetera. Due to the particle-like representation, these methods are usually Lagrangian as the material moves along with the nodes.

This article puts its focus on the SPH (Smoothed Particle Hydrodynamics) as it is one of the most famous meshless methods. **The SPH method** was originally developed in 1977 and has ben greatly developed ever since, and hence **becoming one of the most used nowadays for a variety of purposes, including:**

**Fluid simulations,****Impact analysis,****Fluid-Structure interaction,****Metal forming, etc.**

The following gallery showcases some examples coupling SPH to shell elements in LS-DYNA (click on the gallery to see the animated results). The examples and input decks are available in the following link: http://www.dynaexamples.com/sph and belong to LSTC.

The Smoothed Particle Hydrodynamics (SPH) is a numerical scheme used for solving partial differential equations; originally developed for simulation of astrophysical problems [2; 3]. It was later modified and extended to solid and fluid dynamics problems [4]. It is a meshless Lagrangian method which makes use of interpolation to compute smooth field variables.

In the Finite Element Analysis the domain is discretized into elements whereas **with SPH the domain is divided into particles**, **which are assigned mass, position and velocity, avoiding the classical problems associated with the mesh grid such as mesh distortion** for large deflection problems. However, the use of this scheme also implies some disadvantages as zero energy modes, lack of consistency and tensor instability and the modelling of boundary conditions [5].

The aim of the following sections is to provide a basic understanding of the SPH methodology and cover the basic formulation and parameters required to run this numerical scheme.

First of all, a different formulation had to be derived for the SPH, rather than the one used for typical FEA.

In order to interpolate field variables, instead of a grid and shape functions, a kernel interpolation is used. The value of a function f(x), at an x location, is represented by an integral form of the product of the function and a weighting factor known as the kernel function W(x-x^’,h) [6].

The brackets 〈 〉 symbolize a kernel approximation, h is the so-called smoothing length parameter which defines the domain of the kernel by determining the area of influence of the smoothing function, x’ is the new independent variable and Ω is the domain of the problem.

A better understanding may be acknowledged by looking at the following figure where the kernel function is used to approximate a field variable of a particle i within a radius κh, where κ is a constant applied to the smoothing length.

The integral form of the gradient of the function can be calculated as the following expression:

The kernel function satisfies some inherent properties and requirements [5]:

- Normalization condition:

- Compact support:

The kernel function is zero outside the kernel domainw which depends on the smoothing length κh. These properties ensure that the kernel function behaves as a Dirac delta function when the smoothing length tends to zero [1]:

- Dirac delta function behaviour: , and therefore,

- Kernel approximation properties:

Various options exist to define the kernel function; the most famous ones are the Gaussian and the polynomial splines.

**Gaussian function**

Where the parameter r=(x-x^’)/h is a parameter which makes de domain of the function non-dimensional and d is the number of dimension of the problem (1D, 2D, 3D). This kernel and its derivatives are smooth and stable even for a distorted particle distribution [7].

**Cubic B-spline**

Where C is a scaling factor which assures the compliance with the kernel function properties and depends on the dimension of the problem (e.g. d=1 for 1D):

This spline is used for its narrower support. The B-spline is zero for ranges beyond the kernel domain whereas the Gaussian kernel tends to zero but it is not exactly equal to zero, increasing with the radius of support. However, due to the piecewise definition of the B-spline is less smooth and stable [7].

According to the LS-DYNA theory manual [8] the kernel function implemented in their code is the B-spline. A comparison between these two kernel functions is presented in the following figure.

The integral form of the interpolation can be discretized as the summation of N discrete points:

The value of the filed function for a particle *i* is the addition of the contribution of its neighbouring particles *j* within the interpolation range κh, commonly 2h. The term m_j/ρ_j is the volume associated to each particle *j*. It is important to notice that contrary to the shape functions in FE the value of the function at a node/particle *i* is not its nodal value, it is the contribution of a sum of particles within its neighbouring domain.

The gradient of the function may be written as:

where

and r_ij is the distance between i-j particles.

Once the basic formulation has been covered it can be applied to obtain the conservation equations which govern the particle motion and its thermodynamic state.

A detailed calculation of the derivation process was carried out by Prof. Vignjevic [5] through the use of the kernel previously properties and other mathematical theorems. The final discretized conservation equations are shown below.

Where σ_j^αβ are the components α and β of the stress at a particle j. It is also worth to mention that acceleration vectors, such as gravity, may be summed to the total momentum. Artificial viscosity is also commonly used to improve the numerical stability of the scheme.

Note that the momentum equation may be written in different forms depending on the approximation of choice. E.g In LS-DYNA [8] the following approximations are available: default formulation (0), renormalization approximation (1), symmetric formulation (2), symmetric renormalized approximation (3), tensor formulation (4), fluid particle approximation (5), or fluid particle with renormalization approximation (6). The renormalized approximation refers to a normalized kernel used in order to achieve 1st order consistency. The fluid approximation notorably used to represent fluid behaviour is shown as follows.

A thermodynamic relation between the state variables is needed in addition to the previous set of conservation equations. The EoS can be regarded as a constitutive equation linking the state variables such as temperature, pressure, volume or internal energy in a homogeneous material. It is used to describe the behaviour of fluids, solids and gases. Common relationships between pressure and two other state variables are defined in thermodynamics as:

Where the state variables and parameters are defined as:

Some useful EoS introduced in LS-DYNA [8] and other FE packages are:

**Linear polynomial EoS**Where C_i are user defined constants depending on the material.

**Gruneisen EoS**

Where, the new parameters are defined as follows. C is the speed of sound in the material, S_1,S_2 and S_3 are coefficients of the slope u_s – u_p, γ_0 is a Gruneisen parameter and *a* is the first order volume correction to γ_0.

**Murnaghan EoS**

In its original form, it comes from relating a linear behaviour of the bulk modulus of a solid compressed to a finite strain with respect to the pressure:

Where K_0 is the bulk modulus and K_0′ is its first derivative with respect to the pressure.

**Tabulated EoS**

Where e_V is the natural logarithm of the relative volume, C (Pa) is a constant and γ is the adiabatic index. Introducing up to 10 points, the pressure can be then tabulated and extrapolated if needed.

The smoothing length defines the interaction range between particles. If the smoothing length is set to remain constant the separation between them may become so large that the particles will no longer interact with each other; on the other side, the separation may be too small signficantly slowing down the simulation. In order to avoid this problem a variable smoothing length can be implemented in LS-DYNA [9]:

Where d is the dimension of the problem and div(v) is the divergence of the flow. Additionally, the smoothing length can be limited to vary within specific ranges depending on the initial smoothing length calculated at the beginning of the simulation [9]:

Where HMIN and HMAX are constants applied to the initial smoothing length (h_0) and CLSH is a constant. All these constants can typically be modified.

The neighbour search is one of the main tasks for every SPH time step. It may be a very CPU consuming task depending on the algorithm type selected [5]. Basically two options can be chosen in LS-DYNA:

**Global computation**: every particle is mutually checked with the rest globally. This can lead to a very time-consuming algorithm and it is not recommended for large problems.**Bucket sort**: A grid of size κh is generated and each particle is assigned to one of the boxes, then the searching for each particle starts checking other particles within its own and neighbouring boxes as illustrated below.

Boundary conditions need a special treatment in the SPH method. According to Campbell et al. [10] SPH approximations are not strict interpolants. Hence, the variables in the particle location are not equal to the particle variables and trying to impose a free edge boundary condition to the particles next to the outer surface of a component will lead to the non-satisfaction of the boundary condition (the inner particles will interact with the ones located at the edge). Several approaches have been developed and are summarized below [7]:

- Boundary force method: fixed particles are distributed along the boundaries introducing a force to the inner particles as a function of the distance to the boundary [57].
- Ghost particles – Symmetry planes: a symmetry plane is created and the real particles are mirrored into ghost particles which have the same mass, pressure and velocity as the real ones. This effectively creates a symmetry condition.

- Semi-analytical approaches: there are many semi-analytical approaches covered in the literature. For example Prof. Manenti [7] describes the following where the boundary conditions are approximated through analytical integration terms. Solid boundaries are placed as fluid but unlike the ghost particle method these are an infinite number, producing a continuous boundary.

Contact handling in SPH has been ignored and dealt over the years through the conservation equations implied by the SPH formulation. However, this leads to some penetration when SPH particles from different components interact [10]. A literature survey about this topic reveals some unique SPH contact handling algorithm such as [10] and [11]. The particle to particle contact is commonly used instead of the particle to surface contact due to the difficulties defining a 3D surface out of SPH particles.

Nevertheless, nodes-to-surface contacts have been widely used for SPH – FE coupling. The set of SPH nodes is defined as a slave while the FE master surface is defined as a master. A slave node is simply a nodal point, a SPH particle, whereas a surface is defined using the side of a finite element on the FE component global surface [12].

The most common approach to model contact is the penalty-based method, when a penetration occurs, node-to-node or node-to-surface, a force proportional to the penetration depth is applied to resist and eliminate penetration.

Example: LS-DYNA solves the SPH-FE coupling with a nodes-to-surface penalty based contact, where each SPH slave node is checked for penetration against the master FE surface.

There are many forms of SPH time stepping schemes, all of them explicit schemes such as leap-frog, predictor-corrector method and the low order Runge-Kutta. They have been employed efficiently throughout the years [6].

According to the LS-DYNA theory manual [8], a first order scheme is implemented. The time step varies according to the following formula:

Where C_CFL is a constant and h_i, c_i, v_i are the smoothing length, the adiabatic speed of sound and velocity for a particle i.

Note that when SPH and FE elements coexist for fluid-structure interaction problems, the time-step can be governed by the SPH rather than the FE. This is for example the case when rigid elements are used to describe the structure.

The SPH method have its inherent limitations. A list and brief description of each one is covered below. Prof.Vignjevic [5] described them as follows:

The SPH continuous formulation shows to be inconsistent in its domain, 2h, due to the incompleteness of its kernel support. While in its discrete form, if the particles have an irregular distribution, it loses its 0th order consistency.

The grid must be as regular as possible and without large variations in order to avoid 0th order inconsistency, Mesh 1 above would be preferable to a non-uniform Mesh 2. 1st order consistency may be achieved by a kernel normalisation (CNSPH) leading to a different kernel and conservation equations approximation previously mentioned.

Swegle et al. [13] provided a stability criterion for SPH equations, unstable growth occurs when this condition is reached:Where W^” (x-x’,h) is the second kernel derivative and σ is the stress (σ>0 means tension and σ<0 means compression). This instability arises due to numerical issues from the interaction of the kernel interpolation and constitutive equations changing the nature of the partial differential equations. This results in a clustering of SPH particles which often manifests itself as a tensile mode but it is not restricted to it [5].

Spurious modes, known as zero energy modes, are often found in the FEM and it is not a self-unique feature. They are also present in particle methods such as SPH. There are certain situations when non-physical nodal displacement patterns produce zero strain energy. The main cause being that the field variables and their derivatives are calculated at the particle locations, which for certain field combinations may lead to zero field gradients when interpolated, making these modes easily excitable. Some solutions and further explanation and literature are covered by Prof. Vignjevic [5].

- Mesh-free methods were developed due to inherent problems of the mesh-based approach discretization. SPH can be used to model large deformations avoiding instabilities and extending the simulation time. On the other hand, SPH has limitations of its own such as significant simulation storage and elapsed time consumption. SPH also presents some formulation shortcomings.
- SPH adds a whole new set of variables to the simulation. And hence, a proper understanding of these variables is needed to be reckoned by the analyst.
- Some of the main SPH findings are:
- Field functions are interpolated among the domain where the smoothing length plays a critical role.
- There are different formulations according to the implementation of the momentum equation.
- Boundary conditions are treated differently rather than the usual FE boundary conditions.

- Contact algorithms between FE-SPH are a key feature for fluid-structure interaction and may be modelled in different ways. A surface-to-nodes contact algorithm is widely used in this regard.

**References:**

[1] Liu, G. R. (2003), Mesh free methods: moving beyond the finite element method, 1st ed, CRC Press LLC, Washington D.C.

[2] Lucy, L. B. (1977), “A numerical approach to the testing of the fission hypothesis”, The astronomical journal, vol. 82, no. 12, pp. 1013-1024.

[3] Monaghan, J. J. (1982), “Why particle methods work”, SIAM J. SCI. STAT. COMPUT., vol. 3, no. 4, pp. 422-433.

[4] Monaghan, J. J. (1992), “Smoothed Particle Hydrodynamics”, Annual Review of Astronomy and Astrophysics, vol. 30, pp. 543-574.

[5] Vignjevic, R., ( 2010), Review of Development of the Smooth Particle Hydrodynamics (SPH) Method, School of Engineering, Cranfield University.

[6] Monaghan, J. J. (1988), “An introduction to SPH”, Computer Physics Communications, vol. 48, no. 1, pp. 89-96.

[7] Manenti, S., ( 2009), A Smoothed Particle Hydrodynamics: Basics and Applications, Lecture notes ed., Dipartimento di Meccanica Strutturale, Università degli Studi di Pavia.

[8] Livermore Software Technology Corporation (LSTC) (2006), LS-DYNA Theory manual, Livermore, California.

[9] Livermore Software Technology Corporation (LSTC) (2007), LS-DYNA Keyword user’s manual (v. 971), Livermore, California.

[10] Campbell, J., Vignjevic, R. and Libersky, L. (2000), “A contact algorithm for smoothed particle hydrodynamics”, Computer Methods in Applied Mechanics and Engineering, vol. 184, no. 1, pp. 49-65.

[11] Monaghan, J. J. (1989), “On the problem of penetration in particle methods”, Journal of Computational Physics, vol. 82, no. 1, pp. 1-15.

[12] Attaway, S. W., Heinstein, M. W. and Swegle, J. W. (1994), “Coupling of smooth particle hydrodynamics with the finite element method”, Nuclear Engineering and Design, vol. 150, no. 2–3, pp. 199-205.

[13] Swegle, J. W., Hicks, D. L. and Attaway, S. W. (1995), “Smoothed Particle Hydrodynamics Stability Analysis”, Journal of Computational Physics, vol. 116, no. 1, pp. 123-134.

]]>The FEM (Finite Element Method) is a way of obtaining a of finding a solution to a physical problem. It relies on discretizing a continuum domain into finite elements. The accuracy of the solution greatly depends on the number of elements used to represent the physical domain. As we progressively refine the mesh, the solution improves and given enough iterations it converges towards a specific result. If there is an analytical solution for the given problem, the mesh refinement procedure will converge towards the exact solution.

However, there are situations where the solution does not converge with mesh refinement. Stress singularities are one of these situations. This article puts its focus on stress singularities and stress concentrations. What are they? When do they pose for concern? How should we, as FE analysts, deal with them?

In structural analysis, we are mainly concerned about displacements and their derivatives – the stresses. **A stress singularity is a point of the mesh where the stress does not convergence towards a specific value. As we keep refinement the mesh, the stress at this point keeps increasing**, and increasing, and increasing… Theoretically, **the stress at the singularity is infinite.**

Typical situations where **stress singularities occur are the appliance of a point load, sharp re-entrant corners, corners of bodies in contact and point restraints**. As you can see, stress singularities are a common situation in FEA. The analyst must use his knowledge to determine the possible singularity locations and see if they are of importance for the model or not.

Although stress at these singularities is infinite, this does not mean that the model results are incorrect overall. First of all, the displacements are correct even at the singularity point. On the contrary, the stress at the singularity will pollute the stress results near the singularity, however some distance away from the singularity the stress results will be fine! This is an immediate consequence of St. Venant’s Principle. Perhaps one of the most important principle in FEA as it validates FEA results even with the presence of singularities.

**St. Venant’s principle states that the effect of local disturbances to a uniform stress fields remains local.** Further away from the disturbance the results will not be perturbed. How far away? Typically, as far away as the size of the disturbance.

**St. Venant’s principle allow engineers to dismiss stress singularities when the stress near the singularity is not of interest**. For example:

- Application of loads.

Even if the load is applied as a singular point load creating a stress singularity (σ = P/A and A=0 → σ=∞), the stress distribution away from the applied load will be correct as shown in the image above. Therefore, if we are not interested in the stresses near the singularity point, we should only be concerned about transmitting the resultant load P to the specimen. It does not matter if this is done through a point load or a pressure load, as far as the resultant is transmitted the stresses at the central section will be correct.

As you can see, far away from the point load singularity the stress field is uniform despite of the local singularity. Note that pressure load do not cause point singularities, this is because the pressure attribute is converted by the solver into a set of point loads along the nodes of the edge in a consistent fashion that do not cause a singularity.

- Sharp re-entrant corners.

Sharp re-entrant corners are corners of a mesh which external angle are less than 180º a stress singularity occurs. In the mesh shown below this angle is 90º, an angle of 0º would represent the tip of a crack. In both cases this corner leads to a singularity. No matter how much we refine the mesh, the stresses will not converge at the vicinity of the corner. Nevertheless, St. Venant’s principle states that this singularity only pollutes stress results nearby, allowing us to use stress results away from the corner. The local singularity and its effect is shown in the following image.

The image above is a contour of the global vertical stress over the nominal applied stress at the top. The L structure has been modelled as a plane stress model, typical aluminium properties have been assigned and the right most edge nodes have been restrained. It can be seen that the most compressive and tension stresses both happen to occur at the corner. Therefore, the maximum and minimum stresses of the legend will always increase with mesh refinement!

In reality, no corner can be perfectly sharp. Even if desired, a manufactured sharp corner will always present a small fillet radius. This means that the stress will not be infinite anymore, the corner singularity will disappear. Instead, a stress concentration takes over.

When a large stress gradient occurs in a small, localized area of a structure, the high stress is referred to as a stress concentration [2]

A stress concentration is a place of the mesh where the stress raises above the applied nominal stress. It behaves in a similar fashion than stress singularities, but **the stress will converge towards a finite value**, not infinite, **given that the mesh is sufficiently refined**. Stress concentrations occur when the load path is deviated by the presence of harder or softer features or a change of geometry. For example, such features can be the presence of a hole in a plate mesh, fillet corners, a change of cross-section, etc. Contact forces are another common stress concentrations. High stress gradients also occur near the point of contact and rapidly subside as we move away from the contact area.

Let’s **consider an infinite plate with a small circular hole**. The applied stress is called the nominal stress. Typically the nominal stress is the stress field that would result in the structure without the stress concentration feature – in this case without the hole. As a consequence of the hole, the stress nearby will raise. The radial stress must be zero, there cannot exist stress radially as there is no material in this direction. The increase in stress will be noted in the theta direction, along the perimeter of the hole where the stress keeps increasing with theta, a maximum is found analytically when theta is 90º or 270º. The maximum stress is equal to 3 times the nominal stress. Therefore, for an infinite plate under uniaxial tension with a small hole the ratio between the maximum found stress and the nominal stress is 3. This ratio is the so-called stress concentration factor.

**Stress concentration factor: Kt = σ max / σ nom**

**The nominal stress is calculated using basic elasticity formulas based on the cross-section of the structure.** Sometimes the net section (the section accounting the stress concentration feature) is used, otherwise the gross section is used. Once we know the nominal stress, **the stress concentration factor can be found in tabulated data in the literature.** E.g. [2] and [3]. And the maximum stress can be then retrieved.

A FEA model can be prepared to check this results. A 2D plane stress model of the plate with a hole has been modelled and it is subjected to an arbitrary pressure load at the top edge. Note that only quarter of the model has been modelled, taking advantage of the symmetry conditions. First, a coarse mesh is prepared for the purpose of illustration. Then, the mesh is subsequently refined until the stresses converge towards the theoretical result as shown in the following gallery.

The first mesh shows a stress intensity factor of 2.17. The mesh is too coarse. The second mesh has globally subdivided the element size in a 2×2 fashion. A stress concentration factor of 2.94 is retrieved, almost there! Finally, a 4×4 global subdivision results in a stress intensity factor of 3.03! **Conclusions from this exercise**:

**A coarse mesh will not capture local effects**such as stress concentrations.- The more we refine the mesh, the more accurate the results are. However, the third model is not computationally efficient. St. Venant’s principle says that the effect of the hole within the mesh should be local. Therefore,
**the mesh could be refined locally**(about within 1 diameter away from the hole),**rather****than globally**subdividing all the elements in the mesh. - The final refined mesh shows a stress concentration factor of 3.03 rather than 3, why? Because we have modelled a finite plate rather than an infinite one! When the plate dimensions are finite, a different formula applies which increases a bit the stress concentration factor as expected.

In the previous example FEA has been used to check the mesh convergence towards a stress intensity factor. However, another approach is to **use FEA to calculate stress concentration factors for more complex geometric configurations** which can be later used in engineering hand calculations.

As you can imagine, stress singularities and concentrations are a common occurrence in FEA. So how a FE analyst should deal with them? First of all, the analyst must be aware of potential singularities and cope with them accordingly by one of the following:

**Ignoring the singularities.**If we are interested in the stresses**far away from any singularities, St. Venant’s principle applies – the stresses will be correct**. If we are interested only in displacements or the integrals of the stresses over the singularity region, this magnitudes are also correct at the singularity. A classical operation in FEA is called**“defeaturing”**, many times CAD models have many detailed with features we do not need. E.g. If we are not interested in capturing the stresses at every fillet of the model, we can defeature the model by converting every fillet into sharp corners. This will make the geometry easier to mesh and computationally more inexpensive than a fully featured mesh of the original geometry. We are aware of the singularity, but we choose to dismiss it!**If the stress singularity/concentration at is of importance, then:**- the mesh must be locally refined to capture their effect.
- typical geometric induced singularities such as sharp re-entrant corners can be avoided by modelling fillets instead. Effectively the stress singularity becomes a stress concentration.
- In reality, there is no such thing as infinite stress even at singular points. The material will rather yield or fail before! Thereby,
**rather than using a linear elastic model, we can make use of a elastic-plastic material model**. E.g. a perfect elastic-plastic material will cut-off any stress singularity or concentration past the proportional elastic limit to the yield strength of the material. - In any case,
**the mesh should be refined to verify that stresses do converge**. This requires a**mesh sensitivity study**.

In order to further illustrate how common singularities are in FEA, let’s consider one of the most simple scenarios in the Theory of Elasticity: a cantilever beam under some arbitrary shear force. **When modelling a 2D cantilever beam, there are two singularity points to consider at the top and bottom points of the root section** as shown in the following image. **Why?**

The following are significant beam elasticity theory assumptions for this problem:

- Fiber stress at each section change linearly from the neutral axis to the top/bottom surfaces (σ = M y /I), where
*M*is the bending moment,*y*is the vertical distance from the neutral axis and*I*is the relevant second moment of area. - The normal stress at the top and bottom surfaces must be zero. These edges are free of applied forces.

In a 2D model, the clamped root is under horizontal tensile/compressive stresses. **Due to Poisson’s ratio effect, the cross-section of the beam will tend to deform in the vertical direction**. The top and bottom surfaces are free to do so, however **note that we are imposing a fixity condition at the root in the Y direction**. Suddenly, the top and bottom free surfaces are under the action of a singular point force respectively! The stress normal to the free surfaces jumps from zero stress to infinite stress.

This type of singularity occurs where mixed boundary conditions (displacement and stress boundaries) are applied to a node. On the one hand, the corner nodes must enforce a free edge condition (normal stress = 0). On the other hand, they must restrain the vertical movement of the clamped section. There is no way out but a singularity.

In the next lines, **a mesh sensitivity study** will be carried out. The objective of this study is** to check if the stresses converge given a sufficiently small element size**. A beam element model will be used as a benchmark. Thin beam elements will provide the classical Theory of Elasticity solution.

First of all we should check the displacements. As already mentioned, displacements should not be affected by stress singularities. We expect close agreement between the 2D model and the beam model in that regard (see next image).

The element size of the 2D model was then refined in a 2×2 fashion after each iteration. The tip vertical displacement and fibre stress at the root are extracted and compared to classical beam theory results. The following graph illustrates this process known as mesh sensitivity analysis.

**It is clear that the stresses at the corners do not converge** even with more than 400.000 elements, which equates to less than 0.04% of the beam length!. In fact, **this singularity is of a logarithmic type**, and it is has been thoroughly studied in the literature (see [4] for further reference).

**The up side is that** St. Venant’s principle comes to the rescue. **Stresses far from the singularity are correct. **In the graph below, the fiber and normal stress at the top free surface has been contoured. The stresses vary with the distance from the tip. Normal stresses should be zero, because the edge is not normally loaded. The fiber stresses vary linearly as predicted by beam bending theory. Note however the singularity point at the root. Both fiber and normal stresses suddenly diverge.

Note that if the solution were to be re-run with a Poisson’s ratio of zero, no singularity would occur!

- A mesh sensitivity study is a simple way to check potential singularity points in the mesh.
- Stress singularities do not affect displacement results, not even at the singularities.
- Stress singularities are likely to occur at supports when displacement boundary conditions and stress boundary conditions are mixed. The mesh convergence study shows a logarithmic divergence at the corner of the clamped cantilever beam section.
- The stress results elsewhere are not affected thanks to St. Venant’s principle.
- Beam elements are pretty much underrated compared to 2D and 3D elements. One single beam element was capable of predicting what 400.000 plate elements could not.

- Introducing a elastic-plastic material will cut-off the singular stresses due to yield. For stress concentration locations, Neuber’s rule provides a method to calculate concentration factors past the elastic limit of the material.

**References:**

[1] Saint-Venant’s Principle, Wikipedia, available at: http://es.wikipedia.org/wiki/Principio_de_Saint-Venant (accessed 30 May 2015).

[2] Young, W., Budynas, R., (2001), *Roark’s Formulas for Stress and Strain*, Chapter 17 – Stress Concentrations, McGraw-Hill.

[3] Pilkey, D., Pilkey, W., (2008), *Peterson’s Stress Concentration Factors*, Wiley; 3 edition.

[4] Tullini, N., Savoia, M., (1995) *Logarithmic stress singularities at clamped free corners of a cantilever orthotropic beam under flexure, *University of Bologna.

The Finite Element Analysis, also called the Finite Element Method (FEM), is a numerical technique to find numerical solutions to partial differential or integral equations of field problems. In stress analysis this field is the displacement field whereas in thermal analysis it is the temperature and in fluid flow it is the velocity potential function, and so on [1].

The FEM discretizes the physical domain into subdomains, the so-called finite elements, defined by a set of nodes; the arrangement of elements is called mesh. The unknown field is interpolated by shape functions, usually polynomial functions, depending on a set of discrete variables (i.e. field values at the nodes). By connecting elements the field becomes interpolated over the structure. The unknown field quantity is calculated via minimizing a function such as the potential energy in stress analysis; this procedure leads to a matrix of algebraic equations which are solved with a numerical scheme. As a result, the field values at the nodes are determined and extrapolated over the finite elements [1], [2].

In simple words, **the FEM is a numerical method to solve partial differential equations** by discretizing the domain into a finite mesh. Numerically speaking, **a set of partial differential equations are converted into a set of algebraic equations **to be solved for unknown at the nodes of the mesh. Bear in mind that **FEA is an approximate solution of a mathematical representation of a physical problem**. This is to say that, best case scenario FEA will replicate an analytical solution, whether the mathematical model is according to the real physical situation or not is another matter!

The power of FEA is its versatility. Over other numerical methods many advantages arise:

**applicable to many field problems**: structural analysis, heat transfer, electrical/magnetical analysis, fluid and acoustic analysis, multi-physics, etc.- no geometric restrictions,
- boundary and load conditions are easily applied as well as non-linear characteristics,
- many different materials, models and mathematical representations can be implemented, combined and coupled.

Although FEA has its inherent disadvantages, it has become widely used in structural analysis due to this set of unique features.

As I said, FEA is extensively applied to structural analysis, also called stress analysis, which is my major area of expertise and hence the focus of this blog. Typically, the input and output variables involved in **linear static stress analysis** are the following:

- Unknowns: displacements, strains, stresses.

A finite element solver always solves for displacements at the nodes. Once the displacements are known, other variables can be obtained. Strains are then derived via the compatibility equations from the displacement field, and stresses are subsequently derived from strains via the material constitutive matrix.

- Solid Mechanics equations:
**Equilibrium eqs**. Typically in solid mechanics, any element under static equilibrium satisfies these set of equations.**Constitutive eqs**. This is the relationship between strain and stresses. E.g. For an isotropic material, the Young´s modulus and Poisson’s ratio suffices to describe Hooke’s Law.**Compatibility eqs**. These define how the strains are related to the displacements. Strains are just the derivatives of the displacements.

In total, we need to solve a problem consisting of 15 equations in which 9 are partial differential equations (good luck with that!). For simple cases, analytical solutions are available using the Elasticity theory, but for complex scenarios FEA is required [3].

- Input: Geometry, loading, restraints, material data.

The solid mechanics equations are fine…, but how does FEA solve for displacements?

The FEM uses a variational approach. Let’s consider an elastic body under the action of volume, surface and point loads. **The MPE asserts that the structure shall deform to a position which minimizes the total potential energy of the system**, this principle will allow us to calculate the displacements that satisfy this condition. The Potential energy (PE) of a structural system is defined as the sum of the strain energy (SE) and the work potential (WP)

**MPE applied to a spring-mass system:**

The SE of a mass(m) – spring(k) system is equal to

the WP of a force (f) acting in the X direction is

and therefore the PE is as follows:

If we apply the MPE principle, the PE minimized in the deformed configuration. In order to find this minimum let’s find the first derivative of the PE with respect to x and equate it to zero:

Which gives the position of equilibrium of the mass-spring system under the action of a single force – neglecting the work done by gravity:

**MPE applied to a general elastic body:**

The total potential energy of an elastic body under volume, surface and point forces can be expressed as shown below:

, where epsilon is strain vector and sigma is stress vector, D is the constitutive material matrix relating strain and stress.

, where u is the displacement vector, b is the volume force vector, t is the surface force vector and P is the nodal force vector. Hence, PE is as follows:

We can now apply the compatibility equations, those relating strains and displacements. mentioned before. In doing so, the strain vector can be regarded as a matrix derivative operator, denoted L, times the displacement vector, u. Finally, the PE is written as shown below:

We got the total potential energy of a general elastic body!, but… **where are the finite elements in all this?** Nowhere to be seen, yet… **When the FEM is applied to the MPE the volume and therefore the above equation is discretized into finite elements, within each element the unknown displacements are interpolated using polynomial functions.** In doing so we are assuming that the displacements vary in a polynomial fashion within each element – which may not be correct – this is related to the so-called discretization error. Then, making use of the MPE we will be able to calculate the derivative of the PE with respect to the interpolated displacement field and get the desired solution to our problem.

Firsts things first. For each single element of the discretized finite mesh of our elastic body the MPE applies, and the previous equation ca be re-written as shown below. Note that the displacements have been interpolated using the shape functions and the super-index “*e” *denotes element variables of our discretized domain.

Now that the potential energy has been assessed on an element by element basis, the total potential energy of the system simply adds up to their sum. This is typically called the *assembling process*. The element stiffness matrix, element force and displacement vectors refer to the local element and have to be assembled into a global and much bigger matrix capturing the global stiffness matrix. For example, elements sharing nodes will contribute to the global stiffness matrix together. The stiffness terms coming from each different element is added together to get the total stiffness of the sharing node.

Similarly to the single mass-spring system where the MPE principle results in an algebraic equation, the variational approach produces a set of algebraic equations. The stiffness matrix K and the applied loads vector F will be used then to obtain the unknown nodal displacements.

The process is not yet finished, the restraints have to applied to the global stiffness matrix. As it is, this matrix is singular until the nodal restraints avoiding rigid body motions of the elastic body are set in place. In other words, no static equilibrium of the body can be found until the body is properly restraint. The restraint nodes, whose displacements is known are then removed from the global stiffness matrix. This final reduced stiffness matrix is, somehow, inverted and multiplied by the global force vector. And then, and not before, we will get the displacement vector under the deformed configuration :).

**References:**

Personally, I strongly recommend having a read through the following references which I find very useful to introduce yourself to FEA. Cook’s book very well deserves an honorable mention as it is regarded as the FEA bible. It is a must have on your shelf!

[1] Cook, R. D. (1995), Finite element modeling for stress analysis, 1st ed, John Wiley & sons, INC., University of Wisconsin – Madison.

[2] Fuenmayor, J., Ródenas, J.J., Tarancón, J.E., Tur, M. and Vercher, A., (2009), *Cálculo estructural – Método de los elementos finitos*, Departamento de Ingeniería Mecánica y de Materiales (DIMM), Universidad Politécnica de Valencia.

[3] Qi, H. (2006), What is Finite Element Analysis (FEA)?, University of Colorado, available at: http://www.colorado.edu/MCEN/MCEN4173/chap_01.pdf (accessed 20 May 2015).

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