Isogeometric analysis for solving discontinuous two-phase engineering problems with precise and explicit interface representation

Abstract

This paper presents a new computational method for solving two-phase engineering problems. The novelty of this approach lies in its ability to perform the analysis over the exact two-phase geometry. The geometry is defined by a spline-based representation and constructed based on the level-set method. The level-set-based geometry is mapped into a spline-based geometry using an untrimming technique that is newly developed in this paper. The spline-based representation is defined over unstructured meshes enabling precise representations of geometrically complex domains. An analysis-suitable mechanical model that accurately replicates the geometry is constructed and the governing equations are solved using the isogeometric analysis method. The interface between phases is explicitly represented by cubic B-spline curves, allowing for the evaluation of the solution field and various physical measures over its exact geometry. Additionally, algorithms that allow for changing the continuity of the solution field between the phases while precisely preserving the discontinuous two-phase domain are developed. Consistent analytical sensitivity analysis is formulated for generating the geometrical and mechanical models to allow for solving engineering problems with moving boundaries following the gradient-based approach. The proposed approach is tested for solving Poisson’s equation, linear elasticity, topology optimization involving linear elasticity for compliance minimization, and two-phase flow problems. The results expose the capability of the proposed approach to solve two-phase engineering problems while accurately representing the interface.

Appendices

Appendix 1: Control points locations

This appendix refers to Step 4 in Sect. 2.1, which deals with determining the locations of the control points $\varvec{P}^G$. Since the geometry in this paper is primarily developed based on a level-set function, the locations of $\varvec{P}^G$ are represented as functions of the control parameters of the level-set function $\varvec{\alpha }$, i.e., $\varvec{P}^G(\varvec{\alpha })$. The example in Sect. 2 is used to present the computation procedure. Figure 28 shows the physical domain of $S^0$ with cross marks that are located at the corners of the Bézier patches. Computing $\varvec{P}^G$ starts by determining the values of the level-set function, $\phi$, at these points. This computation is done as follows:

• 1. Compute the level-set function values at the inner corners of the patches in the physical domain. These points are highlighted with blue cross marks in Fig. 28. This computation depends on the level-set function representation. In this paper, the level-set function is represented as a B-spline surface; therefore, the value of the level-set function at a certain point i with the parametric coordinates of $(\xi _i,\eta _i)$ is computed by

  $$\begin{aligned} \phi _i=\sum _{j=1}^{N_{\alpha }} \alpha _j B_j(\xi _i,\eta _i), \end{aligned}$$

  (28)

  where $N_{\alpha }$ is the number of control parameters of the level-set function and $B_j$ is the basis function that corresponds to the control parameter $\alpha _j$

• 2. To avoid untrimming complications along the boundary of the design domain, assign all the evaluation points that are located on the boundary of the design domain with $\phi$ values that correspond to the adjacent layer of evaluation points to the interior of the design domain. For instance, in Fig. 28, all the red cross mark points are assigned with the $\phi$ values that correspond to the blue cross mark points in the adjacent layer.

Fig. 28

Physical domain of $S^0$—demonstrating the evaluation points of the level-set function

Afterward, the control points, $\varvec{P}^G$, of the trimmed patches are located according to the trimming manner across the patch. For each trimmed edge in a trimmed patch, we define a weighting parameter $w_{i-j}$ such as

$$\begin{aligned} w_{i-j}=1-\frac{\phi _i}{\phi _i-\phi _j}, \end{aligned}$$

(29)

where i and j are the indices of the vertices at the end of the trimmed edge. $\phi _i$ and $\phi _j$ represent the values of the level-set function at the vertices i and j. Since this paper deals with quadrilateral faces with four edges that could be trimmed, four weighting parameters could be defined: $w_{1-2}$, $w_{2-3}$, $w_{3-4}$ and $w_{1-4}$ as shown in Fig. 29. If an edge is not trimmed, then the associated weighting parameter is not defined. Note that the direction of computing the weighting parameter is important, while $w_{i-j}=1-w_{j-i}$.

Fig. 29

Demonstrating the weighting parameters $w_{i-j}$

Having the weighting parameters for a trimmed patch, the control points $\varvec{P}^G$ that correspond to the same patch are computed as follows:

$$\begin{aligned} {P_x}^G_i&= \sum _{j=1}^{4} f_{ij}^x(w_{1-2}, w_{2-3},w_{3-4},w_{1-4}) X_j, \end{aligned}$$

(30a)

$$\begin{aligned} {P_y}^G_i&= \sum _{j=1}^{4} f_{ij}^y(w_{1-2}, w_{2-3},w_{3-4},w_{1-4}) Y_j, \end{aligned}$$

(30b)

where

$$\begin{aligned} f_{ij}^x&= a_{ij}^x + b_{ij}^x w_{1-2} + c_{ij}^x w_{2-3} + d_{ij}^x w_{3-4} + e_{ij}^x w_{1-4}; \end{aligned}$$

(31a)

$$\begin{aligned} f_{ij}^y&= a_{ij}^y + b_{ij}^y w_{1-2} + c_{ij}^y w_{2-3} + d_{ij}^y w_{3-4} + e_{ij}^y w_{1-4}; \end{aligned}$$

(31b)

$X_j$ and $Y_j$ are the coordinates of the control points corresponding to the corner of the face. The parameters $a_{ij}^x$, $b_{ij}^x$, $c_{ij}^x$, $d_{ij}^x$, $e_{ij}^x$, $a_{ij}^y$, $b_{ij}^y$, $c_{ij}^y$, $d_{ij}^y$ and $e_{ij}^y$ differ for each case and are chosen in a way that does not cause an overlapping between the faces in the mesh. In the following, we provide the non-zero values of these parameters for each case (Figs. 30, 31, 32, 33):

Fig. 30

Case 1—locating the control points $\varvec{P}^G$

$\underline{{\textbf {Case}}\varvec{1}:}$

$a_{ij}^x$
i	1	2	3	3	3	4	5	5	6	7	7	7	8	9	10	11	12	12	13
j	1	2	1	2	4	2	2	4	2	2	3	4	2	4	4	4	3	4	3
v	1	1	0.5	0.25	0.25	1	0.5	0.5	1	0.33	0.33	0.33	1	1	1	1	0.5	0.5	1

$d_{ij}^x$
i	3	3	5	5	7	7	10	10	11	11	12	12
j	3	4	3	4	3	4	3	4	3	4	3	4
v	0.25	-0.25	0.5	-0.5	0.33	-0.33	0.5	-0.5	1	-1	0.5	-0.5

$a_{ij}^y$
i	1	2	3	3	4	4	5	6	7	8	9	10	11	12	13
j	1	2	1	3	2	3	3	3	3	3	4	3	3	3	3
v	1	1	0.5	0.5	0.5	0.5	1	1	1	1	1	1	1	1	1

$c_{ij}^y$
i	3	3	4	4	5	5	6	6	7	7	8	8
j	2	3	2	3	2	3	2	3	2	3	2	3
v	0.25	$-$ 0.25	0.5	$-$ 0.5	0.5	$-$ 0.5	1	$-$ 1	0.33	$-$ 0.33	0.5	$-$ 0.5

$\underline{{\textbf {Case}}\,\varvec{2}:}$

Fig. 31

Case 2—locating the control points $\varvec{P}^G$

$a_{ij}^x$
i	1	2	3	4	5	6	7	8	9	10
j	1	1	1	1	4	2	2	2	2	3
v	1	1	1	1	1	1	1	1	1	1

$a_{ij}^y$
i	1	2	2	3	4	5	6	7	7	8	9	10
j	1	1	4	4	4	4	2	2	3	3	3	3
v	1	0.5	0.5	1	1	1	1	0.5	0.5	1	1	1

$c_{ij}^y$
i	7	7	8	8	9	9
j	2	3	2	3	2	3
v	0.5	$-$ 0.5	1	$-$ 1	0.5	$-$ 0.5

$e_{ij}^y$
i	2	2	3	3	4	4
j	1	4	1	4	1	4
v	0.5	$-$ 0.5	1	$-$ 1	0.5	$-$ 0.5

$\underline{{\textbf {Case}}\,\varvec{3}:}$

Fig. 32

Case 3—locating the control points $\varvec{P}^G$

$a_{ij}^x$
i	1	2	3	4	5	5	6	7	8	9	10	11	12	13	13	14	15	16	17	17
j	1	2	3	4	1	2	2	2	2	2	2	4	4	3	4	1	1	1	1	2
v	1	1	1	1	0.5	0.5	1	1	1	1	1	1	1	0.5	0.5	1	1	1	0.67	0.33
i	18	18	19	19	19	20	20	20	21	21	21	22	22	23	23	23
j	1	2	1	2	4	1	2	4	1	2	4	2	4	2	3	4
v	0.5	0.5	0.38	0.5	0.12	0.25	0.5	0.25	0.12	0.5	0.38	0.5	0.5	0.33	0.33	0.33

$b_{ij}^x$
i	5	5	6	6	7	7	17	17	18	18	19	19	20	20	21	21
j	1	2	1	2	1	2	1	2	1	2	1	2	1	2	1	2
v	0.5	$-$ 0.5	1	$-$ 1	0.5	$-$ 0.5	0.33	$-$ 0.33	0.5	$-$ 0.5	0.38	$-$ 0.38	0.25	$-$ 0.25	0.12	$-$ 0.12

$d_{ij}^x$
i	11	11	12	12	13	13	19	19	20	20	21	21	22	22	23	23
j	3	4	3	4	3	4	3	4	3	4	3	4	3	4	3	4
v	0.5	$-$ 0.5	1	$-$ 1	0.5	$-$ 0.5	0.12	$-$ 0.12	0.25	$-$ 0.25	0.38	$-$ 0.38	0.5	$-$ 0.5	0.33	$-$ 0.33

$a_{ij}^y$
i	1	2	3	4	5	6	7	8	8	9	10	11	12	13	14	14	15	16	17	17
j	1	2	3	4	1	1	1	2	3	3	3	3	3	3	1	4	4	4	1	4
v	1	1	1	1	1	1	1	0.5	0.5	1	1	1	1	1	0.5	0.5	1	1	0.67	0.33
i	18	18	19	19	19	20	20	20	21	21	21	22	23
j	1	4	1	3	4	1	3	4	1	3	4	3	3
v	0.5	0.5	0.38	0.25	0.38	0.25	0.5	0.25	0.12	0.75	0.12	1	1

$c_{ij}^y$
i	8	8	9	9	10	10	19	19	20	20	21	21	22	22	23	23
j	2	3	2	3	2	3	2	3	2	3	2	3	2	3	2	3
v	0.5	$-$ 0.5	1	$-$ 1	0.5	$-$ 0.5	0.12	$-$ 0.12	0.25	$-$ 0.25	0.38	$-$ 0.38	0.5	$-$ 0.5	0.33	$-$ 0.33

$e_{ij}^y$
i	14	14	15	15	16	16	17	17	18	18	19	19	20	20	21	21
j	1	4	1	4	1	4	1	4	1	4	1	4	1	4	1	4
v	0.5	$-$ 0.5	1	$-$ 1	0.5	$-$ 0.5	0.33	$-$ 0.33	0.5	$-$ 0.5	0.38	$-$ 0.38	0.25	$-$ 0.25	0.12	$-$ 0.12

$\underline{{\textbf {Case}}\,\varvec{4}:}$

Fig. 33

Case 4—locating the control points $\varvec{P}^G$

$a_{ij}^x$
i	1	2	3	4	5	6	7	8	9	10
j	1	1	1	1	4	2	2	2	2	3
v	1	1	1	1	1	1	1	1	1	1

$a_{ij}^y$
i	1	2	2	3	4	5	6	7	7	8	9	10
j	1	1	4	4	4	4	2	2	3	3	3	3
v	1	0.5	0.5	1	1	1	1	0.5	0.5	1	1	1

$e_{ij}^y$
i	2	2	3	3	4	4	7	7	8	8	9	9
j	1	4	1	4	1	4	2	3	2	3	2	3
v	0.5	$-$ 0.5	1	$-$ 1	0.5	$-$ 0.5	0.5	$-$ 0.5	1	$-$ 1	0.5	$-$ 0.5

Appendix 2: Boundary representation

A significant advantage of the proposed approach is its ability to represent explicitly and precisely the boundaries of the phases in the mechanical and the geometrical models as a set of B-spline curves. Corresponding to the bi-cubic spline-based meshes that are used in this paper, all the B-spline boundary curves will be of degree 3. These boundaries are described as periodic curves whose control points and knots vectors are assembled according to the boundary polygons of the unstructured mesh.

We demonstrate the procedure for extracting the boundary curves using the example in Fig. 34. Black and red lines in Fig. 34a, b represent unit and zero knot intervals. The boundary polygons are drawn with thick lines. Boundary control points are colored blue, and all the other control points are colored black. This example has two boundary curves: the inner boundary curve that constitutes the interface between the phases, and the outer boundary curve which is also the boundary of the design domain. We show how these two curves are extracted, while first, we focus on extracting the inner boundary curve.

Initially, the control points and knots intervals of each boundary are sequentially collected in vectors $\varvec{P}^c$ and $\varvec{ed}$. The number of control points in $\varvec{P}^c$ is denoted by N. The order of control points of each boundary curve is depicted in Fig. 34b. For the inner boundary curve, these vectors are

$$\begin{aligned} \varvec{P}^{c}&= [P^{c}_1\;P^{c}_2\;P^{c}_3\;P^{c}_4\;P^{c}_5\;P^{c}_6\; P^{c}_7\;P^{c}_8\;P^{c}_9\;P^{c}_{10}\; P^{c}_{11}\;P^{c}_{12}],\\ \varvec{ed}&= [ed_1\;ed_2\;ed_3\;ed_4\;ed_5\;ed_6\;ed_7\;ed_8\;ed_9\;ed_{10}\;ed_{11}\;ed_{12}]=[1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1]. \end{aligned}$$

Afterward, denoting the vector of control points and the knots vector of a boundary curve by $\tilde{P}^c$ and $\tilde{k}$, respectively, these vectors are assembled according to the following algorithm:

Algorithm 5

The algorithm for assembling the control points and the knots vectors for a single boundary curve.

After applying Algorithm 5 to $\varvec{P}^c$ and $\varvec{ed}$ of the inner boundary polygon, the obtained control points and knots vector of the boundary curve are

$$\begin{aligned} \varvec{\tilde{P}}^{c}&= [P^{c}_1\;P^{c}_2\;P^{c}_3\;P^{c}_4\;P^{c}_5\;P^{c}_6\; P^{c}_7\;P^{c}_8\;P^{c}_9\;P^{c}_{10}\; P^{c}_{11}\;P^{c}_{12}\;P^{c}_1\;P^{c}_2\;P^{c}_3],\\ \varvec{k}&= [-2\;\;-1\;\;0\;\;1\;\;2\;\;3\;\;4\;\;5\;\;6\;\;7\;\;8\;\;9\;\;10\;\;11\;\;12\;\;13\;\;14\;\;15\;\;16]. \end{aligned}$$

Applying the same procedure for the outer boundary polygon, the following vectors are obtained:

$$\begin{aligned} \varvec{P}^{c}&= [P^{c}_1\;P^{c}_2\;P^{c}_3...\;P^{c}_{28}],\\ \varvec{ed}&= [0\;\;1\;\;1\;\;1\;\;1\;\;1\;\;0\;\;0\;\;1\;\;1\;\;1\;\;1\;\;1\;\;0\;\;0\;\;1\;\;1\;\;1\;\;1\;\;1\;\;0\;\;0\;\;1\;\;1\;\;1\;\;1\;\;1\;\;0].\\ \varvec{\tilde{P}}^{c}&= [P^{c}_1\;P^{c}_2\;P^{c}_3...\;P^{c}_{28}\;P^{c}_1\;P^{c}_2\;P^{c}_3]\\ \varvec{k}&= [-1\;\;0\;\;0\;\;0\;\;1\;\;2\;\;3\;\;4\;\;5\;\;5\;\;5\;\;6\;\;7\;\;8\;\;9\;\;10\;\;10\;\;10\;\;11\;\;12\;\;13\;\;14\;\;15\;\;15\;\;15 \;\;16\;\;17\;\;18\;\;19\;\;20\;\;20\;\;20\;\;21\;\;22\;\;23]. \end{aligned}$$

The B-spline boundary curves are highlighted in green on top of the physical domain in Fig. 34c.

Fig. 34

An example to demonstrate the procedure of extracting the boundaries of the phases