# A Two-Dimensional Mathematical Model of Heat Propagation Equations and Their Significance for Soil Temperature

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## Abstract

**:**

## 1. Introduction

## 2. Finite Element Methods

#### 2.1. Finite Element Methods Description

#### 2.1.1. Different Types of d.o.f

#### 2.1.2. Graphical Codification of d.o.f

#### 2.2. Classical ${P}_{K}$ Lagrange Elements on Simplifies

#### 2.3. Other Geometries of Classical Lagrange

#### 2.4. Elements with Hierarchical Basis

#### Hierarchical Elements with Respect to the Degree

#### 2.5. Elements with Hierarchical Basis

#### 2.5.1. Composite Elements

#### 2.5.2. Hierarchical Composite Elements

#### 2.6. Specific Elements in Dimension 1

#### 2.6.1. GaussLobatto Element

#### 2.6.2. Hermite Element

#### 2.6.3. Lagrange Element with an Additional Bubble Function

#### 2.7. Specific Elements in Dimension 2

#### 2.7.1. Elements with Additional Bubble Functions

#### 2.7.2. Non Conforming ${P}_{1}$ Element

#### 2.7.3. Hermite Element

#### 2.7.4. Argyric Element

#### 2.8. Specific Elements in Dimension 3

#### 2.8.1. Elements with additional Bubble Functions

#### 2.8.2. Hermite Element

#### 2.9. Interpolation of Elements on Different Meshes

## 3. Integration Methods

#### 3.1. Integration Methods Description

#### 3.2. Exact Integration Methods

#### 3.3. Newton Cotes Integration Methods

#### 3.4. Gauss Integration Methods on Dimension 1

#### 3.5. Gauss Integration Methods on Dimension 2

## 4. Implementation of the Finite Element Method

#### 4.1. Data Structure of a Mesh

- The list of nodes
- The connectivity table

#### 4.1.1. The List of Nodes

- ${N}_{s}$ number of nodes.
- The coordinates of each node $i=1,...,{N}_{s}$.

- Dimension 1: ${x}_{0}:=a<{x}_{1}<{x}_{2}<...<{x}_{n}:=b$${N}_{s}=n+1$Table with entry of length ${N}_{s}$ (see Figure 40).Do not always store this data structure. If the grid is uniform, it is given implicitly by$${N}_{s}=n+1;{x}_{0}:=a;{x}_{i}:={x}_{0}+ih,i=1,...,{N}_{s}-1$$
- Dimension 2: table of coordinates ${\left\{\left({x}_{1,i},{x}_{2,i}\right)\right\}}_{i=1}^{i={N}_{s}}$Number of nodes: ${N}_{s}$Table with two lines and ${N}_{s}$ columns (see Figure 41).

#### 4.1.2. Connectivity Table

- ${N}_{E}$: number of elements.
- For each element $e=1,...,{N}_{E}$ is given: ${n}_{e,j}$ number of the node j of element e.

- Dimension 1$${N}_{E}={N}_{s}-1$$For each element $e=1,...,{N}_{E}$$${n}_{e,1}:=e,\phantom{\rule{5.69046pt}{0ex}}{n}_{e,2}:=e+1$$For the one-dimensional case, the connectivity table is implicit from the list of vertices. It is therefore useless to store.
- Dimension 2: For a mesh in triangles,${N}_{E}$: number of triangles.${n}_{e,j}$: $e=1,...,{N}_{E}$, $j=1,2,3$

#### 4.2. Meshing and Boundary Conditions

- Dimension 1 : $a:={x}_{0}<...<{x}_{n}:=b$, $\overline{{\mathsf{\Gamma}}_{D}}=\left\{a\right\}$, ${I}_{D}:=\left\{1\right\}$
- Dimension 2 : ${I}_{D}:=\{1,4\}$

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

- Without any restriction if$\mathsf{\Omega}=\left]0,L\right[$
- Under the following angle condition if Ω is a polygonal domain of ${\mathbb{R}}^{2}$$$\left\{\begin{array}{c}{\theta}_{T}\phantom{\rule{4.pt}{0ex}}\mathit{is}\phantom{\rule{4.pt}{0ex}}\mathit{the}\phantom{\rule{4.pt}{0ex}}\mathit{smallest}\phantom{\rule{4.pt}{0ex}}\mathit{angle}\phantom{\rule{4.pt}{0ex}}\mathit{of}\phantom{\rule{4.pt}{0ex}}T\in {\tau}^{h},{\theta}^{h}=\underset{T\in {\tau}^{h}}{min}{\theta}_{T}\hfill \\ {\theta}^{h}\ge {\theta}_{0}>0\phantom{\rule{4.pt}{0ex}}\mathit{independently}\phantom{\rule{4.pt}{0ex}}\mathit{of}\phantom{\rule{4.pt}{0ex}}h\hfill \end{array}\right.$$

#### 4.3. Assembling

#### 4.4. Elementary Matrices

- Elementary Stiffness Matrix$$\underset{{T}^{e}}{\int}\sum _{i,j=1}^{N}{a}_{ij}{\partial}_{j}{u}^{e}{\partial}_{i}{v}^{e}\mathrm{d}{T}^{e}=\left(\begin{array}{ccc}{v}_{1}^{e}& \cdots & {v}_{l}^{e}\end{array}\right){K}^{e}\left(\begin{array}{c}{u}_{1}^{e}\\ \vdots \\ {u}_{l}^{e}\end{array}\right)$$
- Elementary Mass Matrix$$\underset{{T}^{e}}{\int}{a}_{0}{u}^{e}{v}^{e}\mathrm{d}{T}^{e}=\left(\begin{array}{ccc}{v}_{1}^{e}& \cdots & {v}_{l}^{e}\end{array}\right){M}^{e}\left(\begin{array}{c}{u}_{1}^{e}\\ \vdots \\ {u}_{l}^{e}\end{array}\right)$$
- Elementary Volume Loading Matrix$$\underset{{T}^{e}}{\int}f{v}^{e}\mathrm{d}{T}^{e}=\left(\begin{array}{ccc}{v}_{1}^{e}& \cdots & {v}_{l}^{e}\end{array}\right){F}^{e}$$
- Elementary Stiffness Matrix Concentrated on the Edge$$\underset{{\left({G}^{a}\right)}^{{}^{\prime}}}{\int}\lambda {u}^{\left(a\right)}{v}^{\left(a\right)}\mathrm{d}{\left({G}^{a}\right)}^{{}^{\prime}}=\left(\begin{array}{ccc}{v}_{1}^{\left(a\right)}& \cdots & {v}_{{l}^{{}^{\prime}}}^{\left(a\right)}\end{array}\right){K}_{c}^{\left(a\right)}\left(\begin{array}{c}{u}_{1}^{\left(a\right)}\\ \vdots \\ {u}_{{l}^{{}^{\prime}}}^{\left(a\right)}\end{array}\right)$$
- Elementary Loading Matrix concentrated on the Edge$$\underset{{\left({G}^{a}\right)}^{{}^{\prime}}}{\int}h{v}^{e}\mathrm{d}{\left({G}^{a}\right)}^{{}^{\prime}}=\left(\begin{array}{ccc}{v}_{1}^{\left(a\right)}& \cdots & {v}_{{l}^{{}^{\prime}}}^{\left(a\right)}\end{array}\right){F}_{c}^{\left(a\right)}$$Matrices and forces concentrated in dimension $N=1$ are introduced at the initialization or the end of the assembling process.

#### 4.5. Elementary Matrices in One Dimension

- Elementary stiffness matrix$${\left({u}^{e}\right)}^{{}^{\prime}}={\displaystyle \frac{{u}_{2}^{e}-{u}_{1}^{e}}{|{T}^{e}|}},\phantom{\rule{28.45274pt}{0ex}}{\left({v}^{e}\right)}^{{}^{\prime}}={\displaystyle \frac{{v}_{2}^{e}-{v}_{1}^{e}}{|{T}^{e}|}}$$$$\begin{array}{cc}\hfill \left(\begin{array}{cc}{v}_{1}^{e}& {v}_{2}^{e}\end{array}\right){K}^{e}\left(\begin{array}{c}{u}_{1}^{e}\\ {u}_{2}^{e}\end{array}\right)& =\underset{{a}_{1}^{e}}{\overset{{a}_{2}^{e}}{\int}}{a}^{e}{\left({u}^{e}\right)}^{{}^{\prime}}{\left({v}^{e}\right)}^{{}^{\prime}}\mathrm{d}x\hfill \\ & ={\displaystyle \frac{\overline{{a}^{e}}}{|{T}^{e}|}}\left({u}_{2}^{e}-{u}_{1}^{e}\right)\left({v}_{2}^{e}-{v}_{1}^{e}\right)\hfill \end{array}$$$$\overline{{a}^{e}}:={\displaystyle \frac{1}{|{T}^{e}|}}\underset{{a}_{1}^{e}}{\overset{{a}_{2}^{e}}{\int}}{a}^{e}\mathrm{d}{T}^{e}$$$$\overline{{a}^{e}}\approx {\displaystyle \frac{{a}^{e}}{2}}\left({a}_{1}^{e}+{a}_{2}^{e}\right)$$
- Condensed mass matrixGenerally, it is sufficient to calculate the integral (4) in an approximate way by the method of trapezes$$\begin{array}{cc}\hfill \left(\begin{array}{cc}{v}_{1}^{e}& {v}_{2}^{e}\end{array}\right){M}^{e}\left(\begin{array}{c}{u}_{1}^{e}\\ {u}_{2}^{e}\end{array}\right)& =\underset{{a}_{1}^{e}}{\overset{{a}_{2}^{e}}{\int}}{a}_{0}^{e}{u}^{e}{v}^{e}\mathrm{d}x\hfill \\ & ={\displaystyle \frac{|{T}^{e}|}{2}}\left({\left({a}_{0}^{e}\right)}_{1}{u}_{1}^{e}{v}_{1}^{e}+{\left({a}_{0}^{e}\right)}_{2}{u}_{2}^{e}{v}_{2}^{e}\right)\hfill \end{array}$$$${M}^{e}={\displaystyle \frac{|{T}^{e}|}{2}}\left(\begin{array}{cc}{\left({a}_{0}^{e}\right)}_{1}& 0\\ 0& {\left({a}_{0}^{e}\right)}_{2}\end{array}\right)$$
- Elementary loading MatrixHere again we can use the trapeze formula to approximate the integral (5)$$\begin{array}{cc}\hfill \left(\begin{array}{cc}{v}_{1}^{e}& {v}_{2}^{e}\end{array}\right){F}^{e}& =\underset{{a}_{1}^{e}}{\overset{{a}_{2}^{e}}{\int}}{a}_{0}^{e}{f}^{e}{v}^{e}\mathrm{d}x\hfill \\ & \approx {\displaystyle \frac{|{T}^{e}|}{2}}\left({f}_{1}^{e}{v}_{1}^{e}+{f}_{2}^{e}{v}_{2}^{e}\right)\hfill \end{array}$$$${F}^{e}={\displaystyle \frac{|{T}^{e}|}{2}}\left(\begin{array}{c}{f}_{1}^{e}\\ {f}_{2}^{e}\end{array}\right)$$
- Matrices concentrated on the edgeIt is here matrices with a single non-zero coefficient which are given directly from$$\lambda {u}_{{N}_{s}}^{h}{v}_{{N}_{s}}^{h}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}h{v}_{{N}_{s}}^{h}$$$${\left({K}_{c}^{t}\right)}_{{N}_{s}{N}_{s}}=\lambda \phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{\left({F}_{c}^{t}\right)}_{{N}_{s}}=h$$

#### 4.6. Elementary Matrices in Dimension Two

- Elementary stiffness matrixThe following method of determining the elementary matrix is in fact general. She can be used for the determination of any elementary matrix and in the framework of any finite element method. We have by definition$$\left(\begin{array}{ccc}{v}_{1}^{e}& {v}_{2}^{e}& {v}_{3}^{e}\end{array}\right){K}^{e}\left(\begin{array}{c}{u}_{1}^{e}\\ {u}_{2}^{e}\\ {u}_{3}^{e}\end{array}\right)=\underset{{T}^{e}}{\int}\left(\begin{array}{cc}{\partial}_{1}{v}^{e}& {\partial}_{2}{v}^{e}\end{array}\right){A}^{e}\left(\begin{array}{c}{\partial}_{1}{u}^{e}\\ {\partial}_{2}{u}^{e}\end{array}\right)\mathrm{d}{T}^{e}$$Then if we denote by $\overline{{A}^{e}}$ the mean value of ${A}^{e}$ on ${T}^{e}$$$\left(\begin{array}{ccc}{v}_{1}^{e}& {v}_{2}^{e}& {v}_{3}^{e}\end{array}\right){K}^{e}\left(\begin{array}{c}{u}_{1}^{e}\\ {u}_{2}^{e}\\ {u}_{3}^{e}\end{array}\right)=\left(\begin{array}{cc}{\partial}_{1}{v}^{e}& {\partial}_{2}{v}^{e}\end{array}\right)\left(|{T}^{e}|\overline{{A}^{e}}\right)\left(\begin{array}{c}{\partial}_{1}{u}^{e}\\ {\partial}_{2}{u}^{e}\end{array}\right)$$We now express ${u}^{e}$ and ${v}^{e}$ using their polynomial coefficients and a matrix writing$${u}^{e}\left(x\right)={\alpha}_{0}^{e}+{\alpha}_{1}^{e}{x}_{1}+{\alpha}_{2}^{e}{x}_{2}=\left(\begin{array}{ccc}1& {x}_{1}& {x}_{2}\end{array}\right)\left(\begin{array}{c}{\alpha}_{0}^{e}\\ {\alpha}_{1}^{e}\\ {\alpha}_{2}^{e}\end{array}\right)$$$${v}^{e}\left(x\right)={\beta}_{0}^{e}+{\beta}_{1}^{e}{x}_{1}+{\beta}_{2}^{e}{x}_{2}=\left(\begin{array}{ccc}1& {x}_{1}& {x}_{2}\end{array}\right)\left(\begin{array}{c}{\beta}_{0}^{e}\\ {\beta}_{1}^{e}\\ {\beta}_{2}^{e}\end{array}\right)$$We can then calculate$$\left(\begin{array}{c}{\partial}_{1}{u}^{e}\\ {\partial}_{2}{u}^{e}\end{array}\right)=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}{\alpha}_{0}^{e}\\ {\alpha}_{1}^{e}\\ {\alpha}_{2}^{e}\end{array}\right)={B}^{e}\left(\begin{array}{c}{\alpha}_{0}^{e}\\ {\alpha}_{1}^{e}\\ {\alpha}_{2}^{e}\end{array}\right)$$$$\left(\begin{array}{c}{\partial}_{1}{v}^{e}\\ {\partial}_{2}{v}^{e}\end{array}\right)=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}{\beta}_{0}^{e}\\ {\beta}_{1}^{e}\\ {\beta}_{2}^{e}\end{array}\right)={B}^{e}\left(\begin{array}{c}{\beta}_{0}^{e}\\ {\beta}_{1}^{e}\\ {\beta}_{2}^{e}\end{array}\right)$$We then use the relation which makes it possible to connect the polynomial coefficients with the nodal values$$\left\{\begin{array}{c}{\alpha}_{0}^{e}+{\alpha}_{1}^{e}{a}_{11}+{\alpha}_{2}^{e}{a}_{21}^{e}={u}_{1}^{e}\hfill \\ {\alpha}_{0}^{e}+{\alpha}_{1}^{e}{a}_{12}+{\alpha}_{2}^{e}{a}_{22}^{e}={u}_{2}^{e}\hfill \\ {\alpha}_{0}^{e}+{\alpha}_{1}^{e}{a}_{13}+{\alpha}_{2}^{e}{a}_{23}^{e}={u}_{3}^{e}\hfill \end{array}\right.$$$${P}^{e}\left[{\alpha}^{e}\right]=\left[{u}^{e}\right]$$$${P}^{e}=\left(\begin{array}{ccc}1& {a}_{11}^{e}& {a}_{21}^{e}\\ 1& {a}_{12}^{e}& {a}_{22}^{e}\\ 1& {a}_{13}^{e}& {a}_{23}^{e}\end{array}\right),\phantom{\rule{28.45274pt}{0ex}}\left[{\alpha}^{e}\right]=\left(\begin{array}{c}{\alpha}_{0}^{e}\\ {\alpha}_{1}^{e}\\ {\alpha}_{2}^{e}\end{array}\right),\phantom{\rule{28.45274pt}{0ex}}\left[{u}^{e}\right]=\left(\begin{array}{c}{u}_{0}^{e}\\ {u}_{1}^{e}\\ {u}_{2}^{e}\end{array}\right)$$$${P}^{e}\left[{\beta}^{e}\right]=\left[{v}^{e}\right],\phantom{\rule{28.45274pt}{0ex}}\left[{\beta}^{e}\right]=\left(\begin{array}{c}{\beta}_{0}^{e}\\ {\beta}_{1}^{e}\\ {\beta}_{2}^{e}\end{array}\right),\phantom{\rule{28.45274pt}{0ex}}\left[{v}^{e}\right]=\left(\begin{array}{c}{v}_{0}^{e}\\ {v}_{1}^{e}\\ {v}_{2}^{e}\end{array}\right)$$To express the polynomial coefficients using the nodal values, we calculate the inverse ${H}^{e}={\left({P}^{e}\right)}^{-1}$.$$\left[{\alpha}^{e}\right]={H}^{e}\left[{u}^{e}\right],\phantom{\rule{28.45274pt}{0ex}}\left[{\beta}^{e}\right]={H}^{e}\left[{v}^{e}\right]$$This inversion is in fact the determination of the basic functions on the element ${T}^{e}$ of the finite element. As can be seen, this determination can be made (for this element and for any finite element) in a completely implicit way by programming.Using all previous relationships, we get$${\left[{v}^{e}\right]}^{T}{K}^{e}\left[{u}^{e}\right]={\left[{v}^{e}\right]}^{T}{\left({H}^{e}\right)}^{T}{\left({B}^{e}\right)}^{T}\left(|{T}^{e}|\overline{{A}^{e}}\right){B}^{e}{H}^{e}\left[{u}^{e}\right]$$Hence we obtain the elementary stiffness matrix by a simple calculation$${K}^{e}:={\left({H}^{e}\right)}^{T}{\left({B}^{e}\right)}^{T}\left(|{T}^{e}|\overline{{A}^{e}}\right){B}^{e}{H}^{e}$$The mean value of ${A}^{e}$ can be obtained by an approximate integration formula as in dimension one$$\overline{{A}^{e}}\approx {\displaystyle \frac{1}{3}}\left({A}_{1}^{e}+{A}_{2}^{e}+{A}_{3}^{e}\right)\approx {A}_{0}^{e}$$
- Stiffness Matrix concentrated on the edgeWe usually have the data provided by the mesh (otherwise we can build by programming) from the list of edges on the border, each with a reference number that allows you, among other things, to know if the edge is contained in ${\mathsf{\Gamma}}_{N}$ or does not cover ${\mathsf{\Gamma}}_{N}$. We can therefore assume that we have the following data${N}_{a}$: number of edges on the borderAnd for each edge $\left[a\right]=1,...,{N}_{a}$.${n}_{a1}$: number of the first vertex ${a}_{1}^{\left[a\right]}$${n}_{a2}$: number of the second vertex ${a}_{2}^{\left[a\right]}$thus, We can traverse the edges and calculate the elementary matrix giving the contribution to stiffness concentrated on ${\mathsf{\Gamma}}_{N}$$$\left(\begin{array}{cc}{v}_{1}^{\left[a\right]}& {v}_{2}^{\left[a\right]}\end{array}\right){K}_{c}^{\left[a\right]}\left(\begin{array}{c}{u}_{1}^{\left[a\right]}\\ {u}_{2}^{\left[a\right]}\end{array}\right)=\underset{{\left({G}^{\left[a\right]}\right)}^{{}^{\prime}}}{\int}{\lambda}^{\left[a\right]}{u}^{\left[a\right]}{v}^{\left[a\right]}\mathrm{d}{\left({G}^{\left[a\right]}\right)}^{{}^{\prime}}$$$${K}_{c}^{\left[a\right]}={\displaystyle \frac{|{\left({G}^{\left[a\right]}\right)}^{{}^{\prime}}|}{2}}\left(\begin{array}{cc}{\lambda}_{1}^{\left[a\right]}& 0\\ 0& {\lambda}_{2}^{\left[a\right]}\end{array}\right)$$
- Condensed mass matrix.The condensed mass matrix without using the matrix ${H}^{e}$ and by using the integration formula approached from the vertices$$\underset{{E}^{e}}{\int}{a}_{0}^{e}{u}^{e}{v}^{e}\mathrm{d}{T}^{e}={\displaystyle \frac{|{T}^{e}|}{3}}\left({\left({a}_{0}^{e}\right)}_{1}{u}_{1}^{e}{v}_{1}^{e}+{\left({a}_{0}^{e}\right)}_{2}{u}_{2}^{e}{v}_{2}^{e}+{\left({a}_{0}^{e}\right)}_{3}{u}_{3}^{e}{v}_{3}^{e}\right)$$$${M}^{e}:={\displaystyle \frac{|{T}^{e}|}{3}}\left(\begin{array}{ccc}{\left({a}_{0}^{e}\right)}_{1}& 0& 0\\ 0& {\left({a}_{0}^{e}\right)}_{2}& 0\\ 0& 0& {\left({a}_{0}^{e}\right)}_{3}\end{array}\right)$$The fact that the conditional mass matrix is â€‹â€‹diagonal will be decisive in some solving problems in dynamics, that is to say, depending also on the time t.
- Elementary loading MatrixWe have the same as for the elementary loading Matrix of volume in dimension one$${F}^{e}:={\displaystyle \frac{|{T}^{e}|}{3}}\left(\begin{array}{c}{f}_{1}^{e}\\ {f}_{2}^{e}\\ {f}_{3}^{e}\end{array}\right)$$The elementary loading matrix concentrated on the edge is given in the same way by$${F}_{c}^{e}:={\displaystyle \frac{{\left({G}^{\left[a\right]}\right)}^{{}^{\prime}}}{2}}\left(\begin{array}{c}{h}_{1}^{\left[a\right]}\\ {h}_{2}^{\left[a\right]}\end{array}\right)$$
- Assembling. The total stiffness matrix is defined by$$\left(\begin{array}{ccc}{v}_{1}^{h}& \cdots & {v}_{{N}_{s}}^{h}\end{array}\right){K}^{t}\left(\begin{array}{c}{u}_{1}^{h}\\ \vdots \\ {u}_{{N}_{s}}^{h}\end{array}\right)=\underset{\mathsf{\Omega}}{\int}\left(\begin{array}{cc}{\partial}_{1}{v}^{h}& {\partial}_{2}{v}^{h}\end{array}\right)A\left(\begin{array}{c}{\partial}_{1}{u}^{h}\\ {\partial}_{2}{u}^{h}\end{array}\right)\mathrm{d}\mathsf{\Omega},\phantom{\rule{8.5359pt}{0ex}}{u}^{h},{v}^{h}\in {X}^{h}$$Using the decomposition of the integral on $\mathsf{\Omega}$$${\int}_{\mathsf{\Omega}}\{...\}\mathrm{d}\mathsf{\Omega}=\sum _{{T}^{e}\in {T}^{h}}{\int}_{{T}^{e}}\{...\}\mathrm{d}{T}^{e}$$$$\left(\begin{array}{ccc}{v}_{1}^{h}& \cdots & {v}_{{N}_{s}}^{h}\end{array}\right){K}^{t}\left(\begin{array}{c}{u}_{1}^{h}\\ \vdots \\ {u}_{{N}_{s}}^{h}\end{array}\right)=\sum _{{T}^{e}\in {\tau}^{h}}\left(\begin{array}{ccc}{v}_{1}^{e}& \cdots & {v}_{l}^{e}\end{array}\right){K}^{e}\left(\begin{array}{c}{u}_{1}^{e}\\ \vdots \\ {u}_{l}^{e}\end{array}\right)$$We then identify the coefficient of ${u}_{j}^{h}{v}_{i}^{h}$ in both members of this realation to get$${\left[{K}^{t}\right]}_{ij}=\sum _{\{{T}^{e}\in {\tau}^{h},{n}_{\left[e\right]{i}_{e}}=i,{n}_{\left[e\right]{j}_{e}}=j}{\left[{K}^{e}\right]}_{{i}_{e}{j}_{e}}$$Consequences:This last formula summarizes the assembling process. it has several consequences.
- A coefficient ${\left[{K}^{t}\right]}_{ij}$ is null if ${a}_{j}^{h}$ and ${a}_{i}^{h}$ are not the vertices of a same element. In other words, degrees of freedom ${u}_{j}^{h}$ and ${v}_{i}^{h}$ are not coupled (i.e., ${\left[{K}^{t}\right]}_{ij}\ne 0$ or the equation relating to ${v}_{i}^{h}$ depends on the unknown ${u}_{i}^{h}$) only if ${u}_{j}^{h}$ and ${v}_{i}^{h}$ belong to the same element.
- To form ${K}^{t}$, simply browse the elements by adding successively the contribution of each elementary matrix ${\left[{K}^{e}\right]}_{{i}_{e}{j}_{e}}$ to the total matrix ${\left[{K}^{t}\right]}_{ij}$ with $i={n}_{\left[e\right]{i}_{e}}$ , $j={n}_{\left[e\right]{j}_{e}}$.

## 5. Time Dependent Problems

#### 5.1. Evolution Problems Models

#### 5.1.1. Model Problems

- Parabolic problem$$\left\{\begin{array}{c}\left(x,t\right)\to u\left(x,t\right)\hfill \\ \rho \left(x\right){\partial}_{t}u\left(x,t\right)-{\displaystyle \sum _{i=1}^{N}}{\displaystyle \sum _{j=1}^{N}}{\partial}_{{x}_{i}}{a}_{ij}\left(x\right){\partial}_{{x}_{j}}u\left(x,t\right)\hfill \\ +{a}_{0}\left(x\right)u\left(x,t\right)=f\left(x,t\right),\left(x,t\right)\in \mathsf{\Omega}\times \left]0,T\right[\hfill \\ u\left(x,t\right)=0,\phantom{\rule{5.69046pt}{0ex}}\left(x,t\right)\in {\mathsf{\Gamma}}_{D}\times \left]0,T\right[\hfill \\ {\displaystyle \sum _{i=1}^{N}}{\displaystyle \sum _{j=1}^{N}}{n}_{i}\left(x\right){\partial}_{{x}_{i}}{a}_{ij}\left(x\right){\partial}_{{x}_{j}}u\left(x,t\right)=0,\left(x,t\right)\in \mathsf{\Omega}\times \left]0,T\right[\hfill \\ u\left(x,0\right)={u}^{\left(0\right)}\left(x\right),x\in \mathsf{\Omega}\hfill \end{array}\right.$$
- Hyperbolic problem$$\left\{\begin{array}{c}\left(x,t\right)\to u\left(x,t\right)\hfill \\ \rho \left(x\right){\partial}_{t}^{2}u\left(x,t\right)-{\displaystyle \sum _{i=1}^{N}}{\displaystyle \sum _{j=1}^{N}}{\partial}_{{x}_{i}}{a}_{ij}\left(x\right){\partial}_{{x}_{j}}u\left(x,t\right)\hfill \\ +{a}_{0}\left(x\right)u\left(x,t\right)=f\left(x,t\right),\phantom{\rule{8.5359pt}{0ex}}\left(x,t\right)\in \mathsf{\Omega}\times \left]0,T\right[\hfill \\ u\left(x,t\right)=0,\phantom{\rule{5.69046pt}{0ex}}\left(x,t\right)\in {\mathsf{\Gamma}}_{D}\times \left]0,T\right[\hfill \\ {\displaystyle \sum _{i=1}^{N}}{\displaystyle \sum _{j=1}^{N}}{n}_{i}\left(x\right){\partial}_{{x}_{i}}{a}_{ij}\left(x\right){\partial}_{{x}_{j}}u\left(x,t\right)=0,\phantom{\rule{8.5359pt}{0ex}}\left(x,t\right)\in \mathsf{\Omega}\times \left]0,T\right[\hfill \\ u\left(x,0\right)={u}^{\left(0\right)}\left(x\right),\phantom{\rule{8.5359pt}{0ex}}{\partial}_{t}u\left(x,0\right)={u}^{\left(1\right)}\left(x\right),\phantom{\rule{8.5359pt}{0ex}}x\in \mathsf{\Omega}\hfill \end{array}\right.$$

#### 5.1.2. Variational Formulation

#### 5.2. Semi-Discretization in Space

#### 5.2.1. Finite Element Method

- In segments if $\mathsf{\Omega}$ is a segment,
- In triangles if $\mathsf{\Omega}$ is a polygonal domain of the plane,
- in tetrahedrons if $\mathsf{\Omega}$ is a polyhedral domain of space.

#### 5.2.2. Reduction to a Differential System

#### 5.2.3. Matrix Properties of the Differential System

**Proposition**

**3.**

**Proof.**

**Remark**

**1.**

#### 5.3. Time Diagrams

#### 5.3.1. $\theta $-Scheme for the Parabolic Equations

#### 5.3.2. Neumark Scheme

- Order two approximation of the second derivative by a derivation in the sense of centered finite differences$$\begin{array}{c}\left[\ddot{u}\right]\left({t}_{m}\right)\approx {\displaystyle \frac{\left[u\right]\left({t}_{m+1}\right)-2\left[u\right]\left({t}_{m}\right)+\left[u\right]\left({t}_{m-1}\right)}{\mathrm{\Delta}{t}^{2}}}\hfill \\ M{\displaystyle \frac{\left[u\right]\left({t}_{m+1}\right)-2\left[u\right]\left({t}_{m}\right)+\left[u\right]\left({t}_{m-1}\right)}{\mathrm{\Delta}{t}^{2}}}\approx L\left({t}_{m}\right)-K\left[u\right]\left({t}_{m}\right)\hfill \end{array}$$
- Order two approximation for the second member to improve stability properties$$L\left({t}_{m}\right)-K\left[u\right]\left({t}_{m}\right)\approx $$$$\frac{1}{4}}\left(\left(L\left({t}_{m-1}\right)-K\left[u\right]\left({t}_{m-1}\right)\right)+2\left(L\left({t}_{m}\right)-K\left[u\right]\left({t}_{m}\right)\right)+\left(L\left({t}_{m+1}\right)-K\left[u\right]\left({t}_{m+1}\right)\right)\right)$$Then we get the scheme$$\left({\displaystyle \frac{1}{\mathrm{\Delta}{t}^{2}}}M+{\displaystyle \frac{1}{4}}K\right){\left[u\right]}_{m+1}={b}_{m+1}$$$${b}_{m+1}={\displaystyle \frac{1}{4}}\left({\left[L\right]}_{m-1}+2{\left[L\right]}_{m}+{\left[L\right]}_{m+1}\right)+{\displaystyle \frac{1}{\mathrm{\Delta}{t}^{2}}}M\left(2{\left[u\right]}_{m}-{\left[u\right]}_{m-1}\right)-{\displaystyle \frac{1}{4}}K\left(2{\left[u\right]}_{m}-{\left[u\right]}_{m-1}\right)$$We need two initial data: ${\left[u\right]}_{0}$ and ${\left[u\right]}_{1}$, we have$${\left[u\right]}_{0}=\left[{u}^{\left(0\right)}\right]$$let calculate ${\left[u\right]}_{1}$ using Taylor development of order 2$${\left[u\right]}_{1}\approx \left[u\right]\left(\mathrm{\Delta}t\right)\approx \left[u\right]\left(0\right)+\mathrm{\Delta}t\left[\dot{u}\right]\left(0\right)+\left({\displaystyle \frac{\mathrm{\Delta}{t}^{2}}{2}}\right)\left[\ddot{u}\right]\left(0\right)$$$$\left[\dot{u}\right]\left(0\right)=\left[{u}^{\left(1\right)}\right]$$By going back to (16), we can calculate$$\left[\ddot{u}\right]\left(0\right)={M}^{-1}\left({\left[L\right]}_{0}-K{\left[u\right]}_{0}\right)$$

#### 5.4. Modal Analysis

#### 5.4.1. Resolution of a Differential System by Diagonalization

- System of order 1$${v}_{j}\left(t\right)={v}_{j}^{\left(0\right)}exp(-{\lambda}_{j}t)+\underset{0}{\overset{t}{\int}}exp(-{\lambda}_{j}t\left(t-s\right)){F}_{j}\left(s\right)\mathrm{d}s,\phantom{\rule{8.5359pt}{0ex}}j=1,...,N$$
- System of order 2$${v}_{j}\left(t\right)={v}_{j}^{\left(0\right)}cos\left(t\sqrt{{\lambda}_{j}}\right)+{v}_{j}^{\left(1\right)}{\displaystyle \frac{sin\left(\sqrt{{\lambda}_{j}}\right)}{\sqrt{{\lambda}_{j}}}}+\underset{0}{\overset{t}{\int}}{\displaystyle \frac{sin\left(\sqrt{{\lambda}_{j}}\left(t-s\right)\right)}{\sqrt{{\lambda}_{j}}}}{F}_{j}\left(s\right)\mathrm{d}s,\phantom{\rule{8.5359pt}{0ex}}j=1,...,N$$$${v}_{j}\left(t\right)={v}_{j}^{\left(1\right)}t+\underset{0}{\overset{t}{\int}}\left(t-s\right){F}_{j}\left(s\right)\mathrm{d}s$$if ${\lambda}_{j}=0$We come back to the solution w by the inverse transformation of (19)$$w\left(t\right)=Vv\left(t\right)$$

#### 5.4.2. Problem with Generalized Eigenvalues

**Proposition**

**4.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

#### 5.4.3. Modal Resolution

## 6. Application

#### 6.1. Obtaining Initial Data

- The experimental setup are based on the the weather data on 7–10 March 2017. Temperature measurements are taken in in agricultural ground soil at $10,20,30$ and 50 cm levels, where the maximum and minimum values of the temperature are recorded.
- It’s known that for any numerical prediction model, the data at the boundaries of the domain are indispensable. Hence, the $\alpha $ boundary designating the ground temperature is determined for each 10 min through a linear interpolation along the minimum and the maximum values for a given day, while the $\beta $ boundary is determined by referring to a previous work presented in [21]. The heat storage along the soil depth almost deals with the upper 60 cm layer. However, the experimental data show a deeper dependence of the temperature, i.e., depths exceeding 60 cm seem to be sensitive to climatic variations.It ’s possible to estimate the temperature at 60 cm depth in the soil by studying the variation of the thermal gradient (the experimental data at 60 cm depth is not available) and considered to be a constant value for the whole experiment during 24 h (such an assumption is validated in a previous work), and the results are shown in Figure A1.
- Taking $\mathrm{\Delta}t=10min$ and $h=max{h}_{j}=20$ cm .
- The model will be run to get a value each 10 min along 12 h. The obtained results are then compared with the real data in order to estimate the mean squared error.$$EQM=\underset{n=1}{\sum ^{n=N}}\sqrt{\frac{1}{N}{({T}_{\mathrm{expected}}-{T}_{\mathrm{real}})}^{2}}$$
- There are two modes of calculations: The first one so-called with initialization, means that the model can be turned while the data is initialized at 6 h, 12 h and 18 h. The second mode is called without initialization, where the model is executed for the whole deadline without any reset of the data.

#### 6.2. Realistic Experience

#### 6.2.1. Without Initialization

#### 6.2.2. With Initialization

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Without Initialization

#### Appendix A.2. With Initialization

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**Figure 12.**${P}_{K}$ and ${Q}_{K}$ Classical Lagrange element on simplifies, prisms and parallelepiped with hierarchical bases.

**Figure 16.**Hierarchical composition of a ${P}_{K}$ finite element method on a simplex with S subdivisions.

**Figure 21.**${P}_{1}$ Lagrange element on a segment with additional internal bubble function, 3 d.o.f., ${C}^{0}$.

**Figure 23.**Lagrange element on a triangle with additional internal bubble function. (

**a**) ${P}_{1}$ with additional internal bubble function, 4 d.o.f., ${C}^{0}$, (

**b**) ${P}_{2}$ with additional internal bubble function, 7 d.o.f., ${C}^{0}$.

**Figure 24.**${P}_{1}$ Lagrange element on a triangle with additional internal bubble function on face 0, 4 d.o.f., ${C}^{0}$.

**Figure 25.**${P}_{1}$ Lagrange element on a triangle with additional d.o.f. on face 0, 4 d.o.f., ${C}^{0}$.

**Figure 31.**Lagrange element on tetrahedron with additional internal bubble function. (

**a**) ${P}_{1}$ with additional bubble function, 5 d.o.f., ${C}^{0}$. (

**b**) ${P}_{2}$ with additional bubble function, 11 d.o.f., ${C}^{0}$. (

**c**) ${P}_{3}$ with additional bubble function, 21 d.o.f., ${C}^{0}$.

**Figure 32.**${P}^{1}$ Lagrange element on tetrahedron with additional internal bubble function on face 0, 5 d.o.f., ${C}^{0}$.

Type | Expression | Commentary |
---|---|---|

Lagrange type | $\mathsf{\Phi}\left({a}_{i}\right)$ | Value of $\mathsf{\Phi}$ on the node $\left({a}_{i}\right)$. The most simple d.o.f. allows the Lagrange interpolation |

Hierarchical Lagrange Type | $\mathsf{\Phi}\left({a}_{i}\right)\phantom{\rule{3.33333pt}{0ex}}-...$ | Difference between the value of $\mathsf{\Phi}$ on the node $\left({a}_{i}\right)$ and the value of some other base functions. This is generally the bubble functions type of d.o.f. |

Mean type | $\frac{1}{\left|T\right|}{\int}_{T}\mathsf{\Phi}\left(x\right)\phantom{\rule{4pt}{0ex}}dx$ | Value of the mean value of $\mathsf{\Phi}$ on the element. Exists also for the restriction on a face |

Derivative type | $\frac{\partial}{\partial {x}_{i}}\mathsf{\Phi}\left({a}_{i}\right)$ or $\frac{\partial}{\partial \eta}\mathsf{\Phi}\left({a}_{i}\right)$ | Value of a derivative of $\mathsf{\Phi}$ on the node $\left({a}_{i}\right)$. This kind of d.o.f. makes the element no to be $\tau $-equivalent. $\frac{\partial}{\partial \eta}\mathsf{\Phi}\left({a}_{i}\right)$ denotes the normal derivative with respect to a face |

Second derivative | $\frac{{\partial}^{2}}{\partial {x}_{i}\partial {x}_{j}}\mathsf{\Phi}\left({a}_{i}\right)$ | Value of a second derivative of $\mathsf{\Phi}$ on the node $\left({a}_{i}\right)$. This kind of d.o.f. makes also the element no to be $\tau $-equivalent. |

Classical “${\mathit{P}}_{\mathit{K}}$” Lagrange Element “FEM-PK ($\mathit{P},\mathit{K}$)” | ||||||
---|---|---|---|---|---|---|

Degree | Dimension | d.o.f. number | class | vectorial | $\tau $-equivalent | Polynomial |

K, $0\le K\le 255$ | P, $0\le P\le 255$ | $\frac{\left(K+P\right)!}{K!P!}$ | ${C}^{0}$ | ${N}_{0}$, (Q = 1) | Yes $(\tilde{M}=Id)$ | Yes |

Discontinuous “${\mathit{P}}_{\mathit{K}}$” Lagrange Element “FEM-PK-DISCONTINUOUS ($\mathit{P},\mathit{K}$)” | ||||||
---|---|---|---|---|---|---|

Degree | Dimension | d.o.f. number | class | vectorial | $\tau $-equivalent | Polynomial |

K, $0\le K\le 255$ | P, $0\le P\le 255$ | $\frac{\left(K+P\right)!}{K!P!}$ | $Discontinuous$ | ${N}_{0}$, ($Q=1$) | Yes $(\tilde{M}=Id)$ | Yes |

${\mathit{Q}}_{\mathit{K}}$ Lagrange Element on Parallelepipeds “FEM-QK ($\mathit{P},\mathit{K}$)” | ||||||
---|---|---|---|---|---|---|

Degree | Dimension | d.o.f. number | class | vectorial | $\tau $-equivalent | Polynomial |

$KP$, $0\le K\le 255$ | P, $2\le P\le 255$ | ${\left(K+1\right)}^{p}$ | ${C}^{0}$ | ${N}_{0}$, ($Q=1$) | Yes $(\tilde{M}=Id)$ | Yes |

Graphic | Coordinates | Weights | Function to Call/Order | |
---|---|---|---|---|

x | y | |||

1/3 | 1/3 | 1/2 | ”IM-TRIANGLE(1)” 1 point, order 1 | |

1/6 2/3 1/6 | 1/6 1/6 2/3 | 1/6 1/6 1/6 | ”IM-TRIANGLE(2)” 3 points, order 2 | |

1/3 1/5 3/5 1/5 | 1/3 1/5 1/5 3/5 | 27/96 25/96 25/96 25/96 | ”IM-TRIANGLE(3)” 4 points, order 3 | |

a 1−2a a b 1−2b b | a a 1−2a b b 1−2b | c c c d d d | ”IM-TRIANGLE(4)” 6 points, order 4, a = 0.445948490915965, b = 0.091576213509771 c = 0.111690794839005 d = 0.054975871827661 |

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**MDPI and ACS Style**

Dhahbi, A.B.; Boulaaras, S.; Radwan, T.; Mezouar, N.; Zennir, K.; Haiour, M.; Allahem, A.; Ghanem, S.
A Two-Dimensional Mathematical Model of Heat Propagation Equations and Their Significance for Soil Temperature. *Symmetry* **2019**, *11*, 478.
https://doi.org/10.3390/sym11040478

**AMA Style**

Dhahbi AB, Boulaaras S, Radwan T, Mezouar N, Zennir K, Haiour M, Allahem A, Ghanem S.
A Two-Dimensional Mathematical Model of Heat Propagation Equations and Their Significance for Soil Temperature. *Symmetry*. 2019; 11(4):478.
https://doi.org/10.3390/sym11040478

**Chicago/Turabian Style**

Dhahbi, Anis Ben, Salah Boulaaras, Taha Radwan, Nadia Mezouar, Khaled Zennir, Mohamed Haiour, Ali Allahem, and Sewelem Ghanem.
2019. "A Two-Dimensional Mathematical Model of Heat Propagation Equations and Their Significance for Soil Temperature" *Symmetry* 11, no. 4: 478.
https://doi.org/10.3390/sym11040478