FEA project


Oakland University

Question Description

I'm working on a mechanical engineering project and need an explanation to help me study.

have this project and it needs fi, the abaqus was designed and the matlab. the only thing that need to do is bringing your Abaqus mesh to the Matlab and show how this model deforms under the given boundary conditions.

And couple of things. I need one who understands this kind of work very good

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THEORETICAL BACKGROUND In this project, the aim is to assume linear elasticity between the concrete and the soil underneath the gravity dam. Fluids present will be taken as pressure loads and one should also assume that the soil and concrete are perfectly bonded to each other. Therefore, an analysis that’s meant to break down these data into simpler parts, so that it can be reassembled with a supported system of equations to address the problem need to conduct a Finite Element Analysis. It observes a Partial Differential Equations model by nature, and an iterative process. This method allows easier modeling and simulation of complex geometrical and irregular shapes. It allows virtual testing in the sense that specifications can be adjusted should physical properties of a certain project in a design process collide or do not match. Boundary conditions can define which conditions need a response from FEM. Meshes that are generated by the soft wares given for this activities consider different types of elements. Although the problem does not really expect perfect shapes of the figures, assumption of them being geometrically perfect shall be made to support computergenerated calculations done from Abaqus and as well as at the MATLAB. Studying the theoretical background of the Brachistochrone Problem (“Shortest Time”) would be by Galilei back in 1638 who discovered in his last work that a sphere could travel faster from time A to B following that there is a curvature path followed, and the distance traveled also has gravity influencing it. The problem is related with time and delay, and it was only in 1696 when Bernoulli, the father of Variational calculus, was able to solve this due to Jacob’s influence as Bernoulli shows his 15th version in Leipzig. Other scientists and breakthroughs followed, such as from Leibniz in 1697 where he first drafted the discrete variational method with element-wise triangular shape functions; Lagrange where an arbitrary variation, delta y, is now the test function; Courant who introduced triangular and rectangular “finite elements” for the 2D-St.-Venant torsion problem of a prismatic bar (Poisson equation); Isoparametric mapping for quadrilateral elements in 1968; model hierarchies in 1991 that separate errors of discretization and idealization, which is essential for verification, validation, and uncertainty quantification. Until the recent breakthroughs of availability of simulation for FEM which was initially introduced by SimScale in 2013. Many people through time worked on improving this technique of solving problems related to engineering and mathematical models not only because of the method it offers but also one can view the techniques done via simulation. Governing Equations of Gravity Dam The Dam-Reservoir interaction is represented by two coupled differential equations of the second order. The equation of the structure and the reservoir can be written in the form: (1) (2) Where [M], [C] and [K] are mass, damping and stiffness matrices of the structure, and [G], [C’] and [K’] are matrices representing mass, damping and stiffness of the reservoir, respectively. [Q] is the coupling matrices and {f1} is the vector of body force and hydrostatic force. {f2} is the component of the force due to acceleration at the boundaries of dam-reservoir and reservoir foundation. {P} and {u} are the vector of pressure and displacement. is the ground acceleration and ρ is the density of the fluid. The dot represents the time derivative. Plain Strain Plain strain refers to the physical deformation of a body that is characterized by the displacement of material in a direction that is parallel to the given plane. Gravity dam is a solid plain strain structure. Its thickness is much greater than its other two dimensions, that is why it has been analyzed as 2D plain strain structure. Gravity dams are subjected to various forces like hydrostatic pressure, uplift pressure etc. due to which it causes stress concentration within its body. Such stress concentration leads to localized failure zones in the structure. Though the stress concentration is to be localized can leads to drastic damage to important structures like dams. The dam structure failure is often analyzed using tools like Finite Element Method & ABAQUS. NUMERICAL (FEM) BACKGROUND: For this section, the breakdown of some concepts and operations mostly used in this report will be elaborated, such as the derivation of FEM weak forms, Galerkin's method, and discretization by triangular elements, and post processing. Divide and Conquer Approach In the FEM, a continuous domain is discretized into simple geometric shapes called elements. A finite element is a small piece of structure. Nodes appear on the element boundaries and “fasten” the elements together. Step 1 – Discretize the continuum Step 2 – Select interpolation functions Step 3 – Find the element properties Step 4 – Assemble the element equations Step 5 – solve the global equation systems Step 6 – compute additional results (stress, strain, etc.) Derivation of FEM weak forms FEM weak forms are done when breaking down a big component for a project into smaller parts, or preferred as the term derivation from strong to weak forms. For a certain statement to be derived, first, the Ordinary Differential Equation (ODE) is multiplied to a virtual function that satisfies the initial condition. Second, there will be a shifting of derivatives. Third, integration will be done. Fourth would be the application of a boundary condition before arriving at the weak statement of a form. Figure 1. Graphical representation of weak forms Galerkin’s method This method is crucial for the Finite Element Method as this converts a differential equation to a discrete problem. Galerkin’s method involves doing double integrations, apply boundary conditions for the loading, and work on the equation until the deformation or maximum deflections are found. Discretization by triangular elements Discretization is the process of transferring continuous functions, models, variables and equations into discrete counterparts. The process of dividing the body into an equivalent number of finite elements associated with nodes is called as discretization of an element in FEM. Post processing After a finite element model had been prepared and checked, as well as the boundary conditions, this phase now comes next, where investigation of the results of the analysis will be conducted. Errors and warnings will be checked first, then reaction loads at restrained nodes should be summed up. This basically determines whether the solution does not contain any mathematical errors. Boundary conditions based from known physicist has an equivalent function that it can support. For Dirichlet boundary conditions with FEM, it is concerned with displacement. For Neumann, it focuses on imposed strain, stress, and external loads. For Robin boundary conditions, it can work with projects or part of projects with elastic bedding. GIVEN ACTIVITY: Figure 2. Activity Reference PLOTS OF THE TWO MESHES, BOUNDARY CONDITIONS, AND LOADINGS Figure 3. Plot for Abaqus mesh Figure 4. Boundary Condition PLOT THE DEFORMED SHAPE FOR BOTH MESHES ON TWO SEPARATE PLOTS Figure 5. Figure for Plot Shape Figure 6. Boundary Condition Structure PLOT THE Y-DISPLACEMENT ALONG THE LINE ABC FOR BOTH MESHES ON THE SAME FIGURE FOR COMPARISON Figure 7. Fine Mesh Figure Figure 8. MATLAB data PLOT THE VON-MISSES STRESS ON THE DEFORMED CONFIGURATION, ONLY FOR THE FINE MESH Figure 9. Fine Mesh Figure Figure 10. Color Map of the Von Misses contour TWO PRINCIPAL STRESSES FOR EACH ELEMENT IN THE COARSE MESH Figure 11. Boundary Condition on the principal stresses AVERAGE BAND WIDTH OF THE GLOBAL STIFFNESS MATRIX FOR BOTH MESHES Between the Abaqus mesh file and the MATLAB file, Abaqus was able to show the band width effectively and the stiffness matrix through the true-to-scale models and ratio, but MATLAB was able to execute the process even though the codes are all separated from each other, which helps easily determine which code shows a certain sector for the program, like one for the nodal stress, displacements, elasticity graph, and the like. PHYSICAL AND NUMERICAL IMPLICATIONS, INCLUDE CONCLUDING REMARKS Visually and physically, the data and the models shown implied that the concrete is elastic enough when supported by the soil, which are both able to sustain the pressure load exerted by the water, and the dam can withstand it, up to a specific pressure load. When viewing the model and making observations to generate data such as the stiffness matrix, or deflections, Abaqus can effectively aid in providing the visuals. As for the numerical implications, MATLAB is always a handy software to prove theoretical assumptions, as well as mathematical models for topics such as this, especially for Finite Element Analysis, which needs to focus on the data specifically as it is broken down from the start into smaller subsets, only to be reassembled supported with equations that can help in finding out the value of the missing variables. BONUS QUESTION: Implement an isoparametric triangular element and use gauss quadrature for triangles to analyze the problem. Show that the results are in good agreement (should be identical) with the exact integration triangle implemented in the project. When the Gauss quadrature is integrated to the problem, it can prove that the function over a triangular surface can be achieved by computing for the functions at discrete points. It can be that multiplication of the discrete solutions via a weighing function, and then running the discrete values. When Gauss quadrature is applied to triangular areas, it states that the integral of a function, f, over the area can be evaluated as the sum, over n integration points, of the product of the function at each point, a weighting function for each point, and the determinate would be: 𝑛 1 ∫ 𝑓𝑑 𝑆 = ∑ 𝑤𝑖 𝑡𝑖 1𝐽𝑖 2 2 𝑖=1 APPENDIX A The full copies of the codes are in the MATLAB and Abaqus file. For the abaqus code, this represents what would be the near-to-real representation of the figures as well as the mesh. The plots of the graph are represented above. As for the MATLAB code, here we will see the actual process on how the code generates the equations and values necessary for the FEM analysis to occur, such as the Boundary Conditions, Global Assembly, and the Derivations of the Stiffness Matrix. CODE FLOW: 1. Check include file, input files and adjust the values for the input: - Included_flags.m - Input_file_64ele, Input_file_16ele_tension - Input_file_16ele - Input_file_1ele 2. Preprocessor.m 3. Displacements.m, get_stress.m, gauss.m, getsctr.m, BmatElast2D, NmatElast2D 4. Mesh2D 5. Point_and_trac.m 6. Nodal_stress 7. Plotmesh 8. Solvedr.m 9. Stress_contours.m 10. Postprocessor.m 11. elasticity2d.m Finite Element Analysis I (ENME E4332) Fall 2020 Prof. H. Waisman Columbia University DEPT. OF CIVIL ENGINEERING & ENGINEERING MECHANICS Final Project (due 12/21/2020) Stress Analysis of a Gravity Dam on Soil Foundation Consider the concrete built gravity dam shown in the figure. The dam is assumed to be perfectly attached to the soil underneath. We will assume that the soil is fixed on the bottom left end and is free to deform in the x-direction (roller boundary condition) on the bottom right end. The dam holds back the river and hence hydrostatic pressure is developed at the left side of the dam as shown. For simplicity the weight of the dam will be modeled as a uniform pressure load, 𝒲, applied on its top edge. 𝒲 g Note: all units in the figure are in meters. In addition, the following parameters are given: Concrete Soil Water Young Modulus [𝒎𝟐 ] 30 × 109 1 × 109 - Poisson ratio 0.2 0.3 - 2200 1800 1000 𝑵 𝒌𝒈 Density [𝒎𝟑 ] 1 Finite Element Analysis I (ENME E4332) Fall 2020 Prof. H. Waisman Columbia University A 2D finite element analysis is required to make sure that the dam is well designed, assuming a plane strain constitutive law. Assignment: Modify the elasticity MATLAB code posted on courseworks (or write your own new code) to analyze this problem using 3-node CST triangular elements with exact integration. The analysis should include a coarse mesh consisting of 300-500 elements and a fine mesh with 1000-2000 elements. Use Abaqus to generate the mesh for the model, and extract the required element and nodal information needed to generate the same mesh in Matlab. Notes: 1) Abaqus will generate *.inp file that will be available in your working directory, and will include the required information needed for mesh generation. 2) When generating the mesh in Abaqus make sure you have sufficient number of nodes placed along the line ABC where plots are required. 3) Once you are ready to submit the project, zip your MATLAB code (call the file LastName_FirstName_UNI.zip) and upload it to Courseworks. Include a Readme file (Readme.txt file) with the name of your team member (if any), and brief instructions how to run the code. Your code will be tested, therefore be sure that the code can be executed without errors before you zip it, otherwise your project will be considered as incomplete. 4) Include with your zipped code a copy of your report in PDF or Word format. Make sure to use legends, labels, grids and titles in all plots required for the report. The final report (submitted as a hardcopy and also uploaded to courseworks) should include: 1. (10%) Theoretical (physics) Background: e.g. problem statement, governing equations, definition of plane strain and other assumptions used, anything other information you think is important. 2 Finite Element Analysis I (ENME E4332) Fall 2020 Prof. H. Waisman Columbia University 2. (10%) Numerical (FEM) Background: e.g. derivation of FEM weak forms, Galerkin's method, discretization by triangular elements, postprocessing, etc. 3. (10%) Plots of the two meshes, boundary conditions and loadings. Node numbers should only be added to the coarse mesh figure. 4. (20%) Plot the deformed shape for both meshes on two separate plots (include the scaling factor used for the plots – the deformation should be visible). 5. (10%) Plot the y-displacement along the line ABC for both meshes on the same figure for comparison. 6. (10%) Plot the Von-Misses stress on the deformed configuration, only for the fine mesh. Show the points of highest/lowest stresses. Why are these points important? 7. (10%) Compute and plot the two principal stresses for each element in the coarse mesh (plot them as a vector field using arrows plot). Make sure to normalize the arrows in this plot, and in addition paint (or mark) the elements with the maximum principle values. 8. (5%) The code should also output the average band width of the global stiffness matrix (for both meshes). Add it to your report. Is the mesh numbering/connectivity ideal or can one improve it? 9. (5%) Add a short discussion on physical and numerical implications and include some concluding remarks. 10. (10%) Appendix A: add a brief description of the code structure, code flow and its subroutines. The code itself should be uploaded as a zipped file to coarse works. Bonus Question (10%): Implement an isoparametric triangular element and use gauss quadrature for triangles to analyze the problem. Show that the results are in good agreement (should be identical) with the exact integration triangle implemented in the project. Final Remark: It is strongly recommended that you prepare a nice report that may serve you well beyond this course, for instance in job interviews. 3 ...
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