The finite element method (FEM) is one of the most used methods by engineers. It is a necessity for every engineer to understand this method. FEM is now an integral part of most structural analyses. In fact, we not only use FEM in daily analysis, we also use FEM to optimize our structural designs. FEM tools allow us to quickly test many design variations. And let's also optimize our design. What I mean by optimization is a massive reduction in our structures. Nowadays, massive savings in our structures lead to lower product costs, lower transportation costs, etc. FEM is very important for a structural engineer. This saves time, allows for rapid variation of designs, and is often used to make a lightweight product stand out. We will mathematically analyze the FEM and then implement it in Java with the aim of plotting the temperature distribution in the profiles. Say no to plagiarism. Get a tailor-made essay on “Why Violent Video Games Should Not Be Banned”? Get an original essay Calculate and plot the temperature distribution over the area studied. For the 3 cases, we will trace the temperature distribution. Due to the symmetry around the y axis, we may only have to do the calculations for the left or right part, we chose the right part. Since the heat flow only goes in the x direction (on the symmetry axis), we only need to do the calculations for the upper part. The heat flow goes from right to left (the heat flow vector is proportional to the negative temperature gradient). Knowing that the flow is perpendicular to the isotherms, the results below make physical sense. From the temperature distribution graphs, we cannot conclude which profile has the highest thermal resistance. Indeed, thermal resistance is a global parameter and this temperature distribution is local. Calculate the thermal resistivity of the hollow wall for different configurations The purpose of the profile is to insulate. The profile with the highest thermal resistance is by definition the best insulator. The best profile of the 3 cases is case C with a thermal resistance equal to R=2677 K/W. But it is also necessary to take into account the mechanical resistance of the profile. The profile with the most air space is the best insulator (air is a very good insulator compared to the material of the profile) but also the most fragile. It is therefore necessary to find a balance between insulation properties and mechanical resistance properties. To find the best profile, we can vary the dimensions a,b,c and d and store each thermal resistance value in an array. After the calculation, we look for the highest thermal resistance value in this table with the optimal dimensions. . Carry out a convergence study according to the density of the finite element mesh and the calculation time. We check convergence with the resistance value: if the resistance does not change much if we increase the dichotomy, we can conclude that we have reached convergence. We cannot achieve convergence because of the “OutOfMemoryError” error. The reason is that our code is not efficient in terms of memory usage. But in the figures, we can clearly see that the graphs reach a horizontal asymptote. This value of this horizontal asymptote is the converged value of the thermal resistance. Draw the temperature profile inside the cavity wall between two points (for example, between the two sides). This figure represents the temperature profile with a fixed y = 1 for case c. There.