# Integration of Direction Fields with Standard Options in Finite Element Programs

## Abstract

**:**

_{1}= 1 and k

_{2}= 0. In doing so, k

_{1}is parallel to y´, and k

_{2}is oriented perpendicular to this. For this extreme case, it is shown that the isotherms are identical to the solution of y’ = f(x,y). The direction fields, for example, can be velocity vectors in fluid mechanics or principal stress directions in structural mechanics. In the case of the latter, possibilities for application in the construction of fiber-reinforced plastics (FRP) arise, since fiber courses, which follow the local principal stress directions, make use of the superior stiffness and strength of the fibers.

## 1. Introduction

_{2}), the Cu conducting track (k

_{1}) is heated to a constant, relatively high temperature. It can be considered as an isothermal line. The thermal heat flow in the plastic due to conduction is very low. For example, if there are two Cu conducting tracks with constant but different temperatures, then a steady temperature gradient independent of k

_{2}prevails in the plastic between them, even if k

_{2}→0. The Cu-tracks can be described by their position y = y(x), and they control the temperature distribution in the insulator. Alternatively, the position can be described by y’(x) together with their starting positions. If these tracks are infinitely densely distributed, the direction field of the Cu-conductance k

_{1}is described by y´ = f(x,y). Insulation works perpendicular to the Cu-tracks when k

_{2}<< k

_{1}. Should the (infinitely densely) guided conduction paths be directed parallel to an arbitrary direction field y’ = f(x,y), then the following hypothesis shall be mathematically verified:

**Hypothesis:**

_{1}and k

_{2}are tangential to an arbitrarily prescribed direction field y´ = f(x,y) provided that the local orientation of k

_{1}follows the direction field y´, and perfect insulation exists perpendicular to this (k

_{2}= 0).

_{1}/k

_{2}→∞. The following section provides a mathematical verification of the hypothesis set out above. The practicability of the integration method is based on independence from thermal boundary conditions; the isotherms always follow the direction field in nonsingular areas. However, the (weighted) distance of the isotherms from each other is influenced by the boundary conditions and can be controlled by them. In Section 3, the procedure is based on some rules and these are demonstrated practically through examples. The influence of the orthotropy ratio k

_{1}/k

_{2}, as well as the influence of singularities on the course of the isotherms, is investigated.

## 2. The Integration of Direction Fields by Means of the Orthotropic Heat Equation

#### 2.1. The Anisotropic Fourier Heat Conduction Law and the Heat Conduction Equation

**Verification of the hypothesis:**In the two-dimensional case, Fourier’s anisotropic heat conduction law with respect to a Cartesian coordinate system is given by Equation (2), and the steady heat balance without internal heat sources by Equation (3), [11]:

_{1}and k

_{2}being oriented according to the direction field of Equation (1) should be transformed into the Cartesian system. The transformation equations for the 2D heat conduction tensor are analogous to a 2D stress tensor, and can be illustrated using trigonometric relationships with Mohr’s (Stress) Circle, as shown in Figure 2.

_{1}/k

_{2}→∞ and therefore can be validated only for k

_{2}= 0. Setting k

_{2}= 0 in Equation (5) and substituting the simplified expression in Equation (2) we have:

_{2}= 0, results in:

_{T}´. An isotherm is characterized by a constant temperature C. Its equation in implicit form is given by F(x,y) = T(x,y) − C = 0. A directional element of this curve has the slope:

_{x}and q

_{y}cancel out, and the right hand sides in Equations (10a) and (10b) become zero. Back-substitution of (∂T/∂x)/(∂T/∂y) = −f into Equation (10) results in:

_{2}= 0), both heat flux components [q

_{x}, q

_{y}] =

**q**are identical to zero. Therefore, the heat flux components q

_{1}and q

_{2}parallel to the orthotropic (principal) thermal conductivities (k

_{1}, k

_{2}) in Figure 2a are also zero. The resulting isotherms do not contradict their definition; the heat flux along an isotherm is zero and, perpendicular to this, the heat flux is also zero due to k

_{2}= 0, and the equating of Equations (1) and (8) was justified: If the heat fluxes are zero for any direction field y´ = f(x,y), then their divergence div(

**q**) in the form of Equation (3) or (4) is also zero. The latter is the heat conduction equation, which is solved in FE programs for steady-state problems without internal heat sources. From Fourier’s law with the unknowns T and

**q**, a directly solvable equation has been formed for T alone provided that k

_{2}= 0. Whether the orthotropic heat conduction problem in Equation (11) is solved, which is not generally available as a calculation option in FE programs, or Equation (7) is solved by means of an FE program, the same isotherms result and are tangential to the given direction field y´ = f(x,y). Whenever k

_{2}> 0, the proof does not work.

#### 2.2. The Relationship between the Isotherms of Extremely Orthotropic Heat Conduction and the Characteristics of a Partial Linear Differential Equation of the First Order

_{2}= 0) in a simple example, f = y’ = 1 is substituted in Equation (11) and u = 1 in Equation (15). Therefore, both equations are formally identical. The location variable y in Equation (11) corresponds to the time variable t in Equation (15), y and t are interchangeable. By simple integration of Equation (13), dy/dx = f = 1, the isotherms are represented by the straight lines y = x + C. Figure 3 shows the corresponding solution for Equations (11) and (15), which was obtained with the FE program ABAQUS (Dassault Systèmes, Vélizy-Villacoublay, France).

_{0}≤ x ≤ x

_{1}) and T(y = 0, x

_{0}≤ x ≤ x

_{1}) is given by a parabolic temperature distribution. The orientation of k

_{1}is given by the vector (1, 1) in the entire solution area. Extreme orthotropy is achieved by zero-setting of k

_{2}whereby k

_{2}is perpendicular to k

_{1}.

_{1}/(cρ) with finite velocity (c = specific heat, ρ = density). As can be seen from Figure 4, the illustratively plotted 0.01·T

_{max}–isotherms at time t

_{1}< t

_{2}< t

_{3}are not parallel to the direction field y´ = 1. This only applies for the steady-state case when t→∞.

#### 2.3. Temperature Boundary Conditions for Extremely Orthotropic Heat Conduction

**n**=

**q**=

**0**(

**n**= normal vector perpendicular to the boundary). In case of extreme orthotropy when k

_{1}> 0 and k

_{2}= 0, the latter condition is always fulfilled according to Equation (11) both internally as well as on the boundary; therefore, in principle, no further conditions need to be specified. However, without specification of a reference temperature, the FE solution is singular. If a reference temperature T

_{ref}is specified at an arbitrary node, we only obtain the trivial solution T(x,y) = T

_{ref}. Therefore, at least two different temperatures at two different nodes need to be specified in order to obtain a non-trivial solution. All temperature boundary conditions, which are specified at two or more nodes, provide isotherms that are tangential to the given direction field y´ = f(x,y). The difference between a two-node and a multiple-node boundary condition manifests itself through different weightings with respect to the location of the isotherms. In order to obtain appropriate solutions, the direction field must be viewed in connection with the physical task, see Figure 5. This figure shows a disc (half-model) with an elliptical hole under pure bending.

_{1}trajectories, are then distributed equidistantly. For y < 0, T = 0 is defined. The σ

_{2}trajectories are calculated in a further orthotropic heat conduction analysis. These require a linear temperature distribution for y < 0 and T = 0 for y > 0. Both isotherm images are superimposed and result in the two fiber layers in Figure 5a.

_{1}trajectories) and y < 0 (σ

_{2}trajectories) along the right boundary; see Figure 5b.

_{1}trajectories, specification of T(L,h/2) = T

_{max}and T(L,0) = 0 provides similar results, though the isotherms are not distributed equidistantly on the right boundary; see also Section 3.1.

_{1}> 0, k

_{2}= 0), the differential Equation (7) of the second order and the Fourier heat conduction Equation (11) of the first order physically describe the same situation. In Section 2.2, it was shown that the Fourier heat conduction Equation (11) can be traced back to the general linear differential Equation (12). The integration of the corresponding characteristic Equation (13) provides characteristics identical to the isotherms in the Fourier heat conduction Equation (11). The theory of the characteristics also includes the treatment of boundary conditions [14], the rules of which can be applied completely to the Fourier heat conduction Equation (14). Figure 6 shows the most important rule.

## 3. Procedure and Application Examples

- Calculation of the PS directions in all elements (disc, plate, or shell). If this is not provided by the respective FE program used, they can be calculated via the stress components [σ
_{xx}σ_{yy}σ_{xy}]. For example, y´ = dy/dx = tanα = σ_{xy}/(σ_{xx}− σ_{2}); see also Figure 2c. Plates and shells have variable stresses across the thicknesses. The direction field therefore must be evaluated for every “thickness integration point”. - Replacement of the structural elements with thermal elements.
- Transmission of the PS directions (Step 1) to local systems, depending on the FE program used, (ABAQUS: *Orientation).
- Assignment of orthotropic thermal conductivities k
_{1}and k_{2}in these local systems with extreme ratios, e.g., k_{1}/k_{2}> 10^{4}. (k_{2}= 0 is permitted in ABAQUS.) - Definition of thermal boundary conditions according to Section 2.3.
- The ratios k
_{1}/k_{2}> 10^{4}and k_{2}/k_{1}> 10^{4}provide the PS trajectories for the first and for the second principal stress, respectively.

#### 3.1. Influence of the Orthotropy Ratio k_{1}/k_{2} on the Integration Accuracy

_{1}/k

_{2}on the integration accuracy is demonstrated by the following example. Figure 7 shows a perforated disc under axial tension. Due to symmetry, a quarter model is sufficient (U

_{x}= 0, U

_{y}= 0). The supplementary material accompanying this paper includes the ABAQUS input for this example.

_{1}is oriented parallel to the previously calculated PS directions of σ

_{1}. The k

_{2}direction is orthogonal to k

_{1}. Now the ratio k

_{1}/k

_{2}is successively increased, thus the isotherms get more and more tangential to the PS direction of σ

_{1}. The influence of the two-node temperature boundary condition at points P2 and P3 (Figure 7d) on the course of isotherms is no longer perceptible for k

_{1}/k

_{2}> 10

^{4}. Under such simple boundary conditions, it must be ensured that their positions are meaningful. If both temperatures are determined at points P1 and P2, then, practically, two different temperatures are defined on one isotherm.

#### 3.2. The Influence of Singularities on the Course of the Isotherms

_{1}and σ

_{2}are identical. If both principal stresses are zero, then a classic singularity is present. Points with concentrated loads are neither one nor the other. Their isoclines are particularly concentrated at the point of force transmission. What does the course of stress trajectories (represented by isotherms) look like in the vicinity of these points? This can be studied using an example of a circular ring under diametrically opposed single forces. Figure 8a shows the course of the isoclines and stress trajectories that were determined photoelastically by Frocht [15].

_{1}and k

_{2}. Two of these systems are illustratively plotted in the fourth quadrant of Figure 8b. The system k

_{1}= 1, k

_{2}= 0 provides the isotherms as σ

_{1}trajectories. With k

_{1}= 0, k

_{2}= 1, the σ

_{2}trajectories are determined. The calculation of the isoclines is helpful in order to localize the singular points A, D, E, H, and K (Figure 8b, left). The isoclines result from Equation (16), i.e., the contour lines of the stress expression on the right side must be visualized; see also Figure 2c.

_{2}trajectories (with σ

_{2}< σ

_{1}). A uniform distribution is achieved through a linearly increasing temperature along RF, with continuation along JL.

_{2}trajectory along RF turns into the σ

_{1}trajectory at point F along FJ. The FJ boundary segment itself is an isotherm and no variable temperature specification is allowed there.

_{1}trajectories. The temperature is defined as linearly increasing along PB with continuation along LN.

_{1}trajectories along PB; however, these end up unevenly distributed on the PR boundary.

#### 3.3. The Principal Stress Trajectories in Shell Structures

_{0}: T = 0 °C, Q

_{1}: T = 1 °C in Figure 10a, a better fiber distribution can be attained than is depicted in Figure 1a. This temperature distribution, shown in Figure 10a, provides the approximate position of three local temperature extremes at the positions P

_{0}, P

_{1}and P

_{2}.

_{0}: T = 0 °C, P

_{1}: T = 1 °C, P

_{2}: T = 2 °C, Figure 10b), the uniformity of the isotherms is improved. A specification of the linear varying temperatures along boundary segments analogous to the previous section would yield further improvement. However, in light of the result in Figure 10b, it does not appear to be crucial.

#### 3.4. Optimization of Fiber Placement in FRP Constructions

_{1}and σ

_{2}are perpendicular to each other, forming a curvilinear cross-ply (CCP) laminate. With regard to optimality, attention must be paid to the algebraic signs of principal stresses.

_{1}direction. The direction field y´ for the largest principal stress σ

_{1}is then represented by two direction fields: y´ + β and y´ − β, as shown in Figure 11.

_{1}·σ

_{2}< 0 (CCP-laminate is optimal) and σ

_{1}·σ

_{2}> 0 (CBAP-laminate is optimal). In the latter case the modified direction fields can also be integrated without restriction by means of the orthotropic heat conduction analysis. The correction angle β is dependent on the material and the principal stress ratio σ

_{1}/σ

_{2}[17]. It should be noted that the orthotropic heat conduction is suitable for the integration of arbitrary direction fields even if they are modified for reasons determined by the engineer. Reference is made to Moldenhauer [1] (pp. 54–60] and [18] for further details and consideration of particular aspects of load-related optimization (layer thickness distribution, loading condition range, and nonlinearity of stress distribution over the shell thickness).

## 4. Discussion and Conclusions

- The integration of the direction fields can take place with standard FE programs with options for an orthotropic heat conduction analysis. No additional programming is necessary.
- Any number of isotherms can be extracted from the integrated continuous temperature field.
- Quick and accurate results are obtained with a simple two-node temperature boundary condition; however, the distribution of isotherms may be non-uniform if singular points are present in the direction field.
- As the computed temperature field is continuous, valuable additional information can be extracted from this field. For example, the density of the isotherms (tensor lines) can be computed by displaying the temperature gradients, see [1] (p. 96).
- Applying the proposed integration method to stress tensors it should be noted that principal stress direction fields can originate from linear or nonlinear analyses, see [1] (p. 74). If principal shear stress (or strain) directions are evaluated in the plastic range then the trajectories can be regarded as slip-lines, see [1] (pp. 41–44).
- Civil engineering: Aligning reinforcement in concrete structures parallel to PS directions is a meaningful tool to effectively increase the low tension strength of concrete, see also [1] (p. 78).

_{xx}σ

_{yy}σ

_{xy}] over the thickness produces normal forces

**N**= [N

_{xx}N

_{yy}N

_{xy}] and bending moments

**M**= [M

_{xx}M

_{yy}M

_{xy}]. FE programs generally provide these section forces and moments

**N**and

**M**, which are both constant across the thickness. Their principal directions (1,2) can be computed in analogy to Equation (16), i.e., y

_{1}´ = N

_{xy}/(N

_{xx}− N

_{2}), y

_{2}´ = −1/y

_{1}´. It makes sense to arrange the M-layers on the inside and outside and the N-layers in the middle of the shell. Symmetric layers are then obtained with [M

_{1}/M

_{2}/N

_{1}/N

_{2}]

_{S}, see also [1] (pp. 64–73).

**q**= k∙grad(T) with k = k

_{1}= k

_{2}. For extreme orthotropy (k

_{1}/k

_{2}→∞) we have an orthotropic heat flux, which is zero everywhere, see Equation (11). If the temperature field is based on isotropic conduction with constant k, then the isotropic heat flux is the correct gradient. For this reason, the nodal temperatures from the orthotropic analysis must be used as prescribed nodal temperatures in an additional thermal run. More details can be found in [1] (p. 96).

## Supplementary Materials

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Shell under lateral pressure p = 0.15 MPa, bedded elastically (K

_{f}= 1000 N/mm); typical dimensions Δx, Δy, and Δz, modulus E = 50 GPa, thickness t = 2 mm. (

**a**) stress trajectories, standard integration begins in the red start elements; (

**b**) PS directions from a linear static analysis; (

**c**) detail, standard integration begins in element A. Improved accuracy is gained with mesh refinement.

**Figure 2.**Transformation between local and global conductivity tensors

**k**in the temperature field T(x,y): (

**a**) Angle α and (π/2) − α between the local system (1,2) and the global x-axis,

**q**= heat flux density; (

**b**) Mohr’s Circle for k

_{1}> 0 and k

_{2}= 0; (

**c**) Mohr’s Circle for k

_{1}> 0 and k

_{2}> 0 by analogy with Mohr’s Stress Circle.

**Figure 3.**The Fourier heat conduction equation with k

_{1}/k

_{2}→∞ is hyperbolic and has transport character. The diagram is valid for the x-y plane as well as for the x-t plane. As an example, the direction field y´ = f(x,y) = 1 is integrated with the orthotropic heat conduction (k

_{2}= 0).

**Figure 4.**The steady-state problem from Figure 3 analyzed in the time domain. The isotherms are parallel to the given direction field y´ = 1 only for the final steady state when t→∞.

**Figure 5.**FRP disc with hole under bending: (

**a**) Linearly increasing temperature boundary conditions for σ

_{1}and σ

_{2}trajectories yield a uniform distribution of fibers (isotherms); (

**b**) parabolic temperature boundary conditions yield a weighted distribution of fibers, which carries a bending moment more effectively.

**Figure 6.**The solution curves T(x,y) of the direction field y´ = f(x,y) are simultaneously the characteristics of the Fourier heat conduction Equation (11). The specification of arbitrary boundary conditions along P2P3 is not permitted since this curve is a characteristic or isoline (y = const.).

**Figure 7.**Perforated disc under axial tension and calculation of the PS lines (isotherms) dependent on heat conduction orthotropy k

_{1}/k

_{2}, where the orientation of k

_{1}and k

_{2}corresponds to the PS directions σ

_{1}and σ

_{2}: (

**a**) k

_{1}/k

_{2}= 1; (

**b**) k

_{1}/k

_{2}= 16; (

**c**) k

_{1}/k

_{2}= 128; (

**d**) k

_{1}/k

_{2}= 4096; (

**e**) algebraic signs of the two principal stresses; this point is addressed in Section 3.4.

**Figure 8.**Circular ring under diametrically opposed single forces, r

_{2}/r

_{1}= 2: (

**a**) Photoelastic isoclines (left) and trajectories (right) [15]; (

**b**) Verification with ABAQUS, isoclines (left), σ

_{1}trajectories and σ

_{2}trajectories (right).

**Figure 9.**Circular ring analogous to Figure 8: (

**a**) Determination of the σ

_{2}trajectory through specification of a linearly increasing temperature along RF and JL; (

**b**) Simplified analysis with a two-node temperature boundary condition; T

_{0}and T

_{L}for the σ

_{1}trajectories, T

_{0}and T

_{L}for the σ

_{2}trajectories.

**Figure 10.**Shell structure analogous to Figure 1: (

**a**) Isotherms with a two-node temperature boundary condition at Q

_{0}and Q

_{1}; (

**b**) Isotherms with a three-node temperature boundary condition at P

_{0,}P

_{1}, P

_{2}, right: Temperature field for extraction of any number of isotherms (fibers).

**Figure 11.**Perforated disc (detail) similar to Figure 7e. In areas with the same principal stress algebraic sign, the CBAP-laminate is optimal: (

**a**) Correction angle ±ß regarding the σ

_{1}direction; (

**b**) Integration of the modified direction field.

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Moldenhauer, H. Integration of Direction Fields with Standard Options in Finite Element Programs. *Math. Comput. Appl.* **2018**, *23*, 24.
https://doi.org/10.3390/mca23020024

**AMA Style**

Moldenhauer H. Integration of Direction Fields with Standard Options in Finite Element Programs. *Mathematical and Computational Applications*. 2018; 23(2):24.
https://doi.org/10.3390/mca23020024

**Chicago/Turabian Style**

Moldenhauer, Herbert. 2018. "Integration of Direction Fields with Standard Options in Finite Element Programs" *Mathematical and Computational Applications* 23, no. 2: 24.
https://doi.org/10.3390/mca23020024