Materials Science & Engineering University of Michigan Summer School for Integrated Computational Materials Education 2018 Computational Mechanics and Finite Element Method Katsuyo Thornton Materials Science & Engineering University of Michigan Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Topics What is FEM? What is FEM good for? Stiffness Method for Mechanical Equilibrium What goes into the method How it’s solved More General Approach Acknowledgement: a few slides were adopted from Edwin Garcia’s lecture in 2017. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Purpose: Learn what Finite Element Method is and what mathematics & physics it is based on. Motivation: It’s the method behind most widely used tools (such as ANSYS/Abaqus/Fluent, COMSOL, etc.). Understanding the method can guide you to use the tools correctly . Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

What is FEM? The finite element method is a numerical method for solving problems of engineering and physics. Useful for problems with complicated geometries, loadings, and material properties where analytical solutions cannot be obtained. Mathematically, the PDE is converted to its variational (integral) form. An approximate solution is given by a linear combination of interpolation functions. The solution is given by error reduction. Physically, it is equivalent to dividing up a system into smaller pieces (elements) where each piece follow the law of nature.

Discretizations Model a physical body by dividing it into an equivalent system of smaller bodies or units (finite elements) interconnected at points common to two or more elements (nodes or nodal points) and/or boundary lines and/or surfaces. BIG PICTURE: Relate the nodal values of the solutions to each other and known values (boundary conditions). A PDE turns into a matrix equation, which can then be numerically solved. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Element Types Felippa C., FEM Modeling: Introduction http://caswww.colorado.edu/courses.d/IFEM.d/IFEM.Ch06.d/IFEM.Ch06.pdf Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Typical Application of FEM Structural/Stress Analysis Fluid Flow Heat Transfer Electro-Magnetic Fields Soil Mechanics Acoustics Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Advantages Irregular Boundaries General Loads Different Materials Boundary Conditions Variable Element Size Easy Modification Dynamics Nonlinear Problems (Geometric or Material) Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Steps in FEM Discretize and Select Element Type Select a Displacement Function Define Strain/Displacement and Stress/Strain Relationships Derive Element Stiffness Matrix & Eqs. Assemble Equations and Introduce B.C.’s Solve for the Unknown Displacements Solve for Element Stresses and Strains Interpret the Results Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Stiffness Method A physically based FEM Divide up a system into smaller pieces (elements) where each piece follow the law of nature Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Definitions for this section For an element, a stiffness matrix is a matrix such that where relates local coordinates and nodal displacements to local forces of a single element. Bold denotes vector/matrices. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Spring Element k 1 2 L Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Definitions node node k - spring constant These are scalar values Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Stiffness Relationship for a Spring Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Steps in Process Discretize and Select Element Type Select a Displacement Function Define Strain/Displacement and Stress/Strain Relationships Derive Element Stiffness Matrix & Eqs. Assemble Equations and Introduce B.C.’s Solve for the Unknowns (Displacements) Solve for Element Stresses and Strains Interpret the Results Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Step 1 - Select the Element Type k 1 2 T T L Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Step 2 - Select a Displacement Function This is a function, not a nodal value. Assume a displacement function Assume a linear function. Number of coefficients = number of d-o-f Write in matrix form. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Express as function of and Solve for a2 : Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Substituting back into: Yields: Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

In matrix form: Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Here, the bases are linear functions! Shape Functions N1 and N2 are called Shape Functions or Interpolation Functions. They express the shape of the assumed displacements. N1 =1 N2 =0 at node 1 N1 =0 N2 =1 at node 2 N1 + N2 =1 Recall Fourier Transform, in which the basis functions are sinusoidal functions. Here, the bases are linear functions! Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

N1 1 2 L Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

N2 1 2 L Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

N1 N2 1 2 L Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Step 3 - Define Strain/Displacement and Stress/Strain Relationships Here is where physics comes into play! T - tensile force  - total elongation Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Step 4 - Derive the Element Stiffness Matrix and Equations Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Stiffness Matrix Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Note: not simple addition! An example later. Step 5 - Assemble the Element Equations to Obtain the Global Equations and Introduce the B.C. (e) indicates “element” index Note: not simple addition! An example later. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Step 6 - Solve for Nodal Displacements Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Step 7 - Solve for Element Forces Once displacements at each node are known, then substitute back into element stiffness equations to obtain element nodal forces. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Two Spring Assembly 2 1 3 k1 k2 2 1 x F3x F2x Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Continuity/Compatibility Condition Elements 1 and 2 remain connected at node 3. This is called the continuity or compatibility requirement. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

This is just adding. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Substitution will give you these equations Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

But, this way is really cumbersome! Computers can’t do this either! Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Assembly of [K] - An Alternative Method 2 1 2 1 3 x F3x F2x k1 k2 Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Recall that Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Assembly of [K] - An Alternative Method 1 3 Insert row and column 2 with zeros Flip row, flip columns, and insert row 1 with zeros 3 2 Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Expand Local [k] matrices to Global Size Computers can do this! Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Net Force Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Compatibility Displacements of the shared nodes are equal. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Boundary Conditions Must Specify B.C.’s to prohibit rigid body motion. Two type of B.C.’s Homogeneous - displacements = 0 Nonhomogeneous - displacements = nonzero value Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Delete row and column corresponding to B.C. Homogeneous B.C.’s Delete row and column corresponding to B.C. Solve for unknown displacements. Compute unknown forces (reactions) from original (unmodified) stiffness matrix. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

2 1 2 1 3 x F3x F2x k1 k2 Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Example: Homogeneous BC, d1x=0 Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Turn the Continuum Equations into Algebraic Equations More General Approach Turn the Continuum Equations into Algebraic Equations Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Spatial Discretization of an Elastic Solid

Another Example: Heat Equation Consider steady-state heat equation with source term and spatially dependent material properties: COMSOL solution for steady-state heat transfer

General Approaches of Finite Element Method Define a set of piecewise polynomial functions (related to the element types and shape functions). The solution can be represented by this piecewise polynomial function. The differential equation can then be integrated over each element with the approximate solution, leading to a set of algebraic equations (key!). The computational mesh no longer needs to be Cartesian or regularly ordered, a primary advantage of the FEM. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Solution is Represented as a Sum of Basis Functions Solution for u is represented by sum of the basis functions times constant coefficients. In FEM, the basis function is called the shape function, and nonzero only on one element. At each point, the coefficients of the polynomial are stored Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

In 1D Linear elements use linear basis functions This is equivalent to piecewise linear approximation. Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

In 2D 7 8 9 6 1 2 3 4 5

For Mechanics: Energy Balance stored energy = applied external loads stored elastic energy work performed by gravity externally applied traction

Application to Heat Equation Consider steady-state heat equation with source term and spatially dependent material properties: The solution is represented by The residual (how different the approximated solution is from the true solution) is given by: Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

The Method of Weighted Residuals (Specifically, Galerkin) The method of weighted residuals requires the weighted integrals of the residuals to be zero. Weight function, wi(x)=φi (x) (Galerkin) Residual, Weighted residual (WR): L1 and L2 are the location of the domain boundaries. i i Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

MWR, continued Consider weight functions, For each ϕi(x), set WR=0: This provides N equations that can be used to solve for in Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Summary & Conclusion We covered a high-level overview of FEM, focusing on the physical model (the stiffness method), advantages & applications. FEM is a deep, complex approach. Some textbooks are in three volumes! It can take an entire semester to cover introduction to FEM! Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Summary & Conclusion We covered a high-level overview of FEM, focusing on the physical model (the stiffness method), advantages & applications. FEM is a deep, complex approach. Some textbooks are in three volumes! It can take an entire semester to cover introduction to FEM! Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018

Example of Software Finite Element Software (focused applications) Fluent (fluid dynamics) ANSYS Mechanical, Structural, ABAQUS, etc. Finite Element Software (multiphysics) With various modules: heat transfer, mechanics, etc. COMSOL, ANSYS, SIMULIA, etc. Free: OOF, FreeFEM++, FEniCS, etc. See https://en.wikipedia.org/wiki/List_of_finite_element_software_packages

Questions? Summer School for Integrated Computational Materials Education University of Michigan, June 4-15, 2018