A Study on Adaptive Implicit–Explicit and Explicit–Explicit Time Integration Procedures for Wave Propagation Analyses
<p>(<b>a</b>) Time interpolation and (<b>b</b>) computational flowchart for the sub-cycling process.</p> "> Figure 2
<p>Spectral radii for the discussed solution procedure (Equation (4a,b)), considering the γ parameter defined by Equation (5b) (implicit approach) and the α parameter defined by (<b>a</b>) Equation (6b) and (<b>b</b>) Equation (6c), for <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </msubsup> </mrow> </semantics></math> = 2.0, 2.1, …, 3.5 (lighter to darker gray color). Results for the CD and the TR are also depicted as black dotted and dashed lines, respectively, for reference.</p> "> Figure 3
<p>Spectral radii for the discussed solution procedure (Equation (4a,b)), considering the γ parameter defined by Equation (5a) (explicit approach) and the α parameter defined by (<b>a</b>) Equation (6b) and (<b>b</b>) Equation (6c), for <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </msubsup> </mrow> </semantics></math> = 0.5, 0.6, …, 2.0 (lighter to darker gray color). Results for the CD and the TR are also depicted as black dotted and dashed lines, respectively, for reference.</p> "> Figure 4
<p>Period elongation and amplitude decay errors for the discussed solution procedure (Equation (4a,b)), considering the γ parameter defined by Equation (5b) (implicit approach) and the α parameter defined by (<b>a</b>) Equation (6b) and (<b>b</b>) Equation (6c), for <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </msubsup> </mrow> </semantics></math> = 2.0, 2.1, …, 3.5 (lighter to darker gray color). Results for the CD and the TR are as well depicted as black dotted and dashed lines, respectively, for reference.</p> "> Figure 4 Cont.
<p>Period elongation and amplitude decay errors for the discussed solution procedure (Equation (4a,b)), considering the γ parameter defined by Equation (5b) (implicit approach) and the α parameter defined by (<b>a</b>) Equation (6b) and (<b>b</b>) Equation (6c), for <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </msubsup> </mrow> </semantics></math> = 2.0, 2.1, …, 3.5 (lighter to darker gray color). Results for the CD and the TR are as well depicted as black dotted and dashed lines, respectively, for reference.</p> "> Figure 5
<p>Period elongation and amplitude decay errors for the discussed solution procedure (Equation (4a,b)), considering the γ parameter defined by Equation (5a) (explicit approach) and the α parameter defined by (<b>a</b>) Equation (6b) and (<b>b</b>) Equation (6c), for <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </msubsup> </mrow> </semantics></math> = 0.5, 0.6, …, 2.0 (lighter to darker gray color). Results for the CD and the TR are also depicted as black dotted and dashed lines, respectively, for reference.</p> "> Figure 6
<p>Adopted spatial discretizations for the first example: (<b>a</b>) discretization 1 (50 k elements); (<b>b</b>) discretization 2 (100 k elements); (<b>c</b>) discretization 3 (150 k elements); and (<b>d</b>) discretization 4 (200k elements).</p> "> Figure 7
<p>Computed values for (1) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </msubsup> </mrow> </semantics></math> and (2)<math display="inline"><semantics> <mrow> <mtext> </mtext> <msubsup> <mrow> <mi mathvariant="sans-serif">γ</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mi mathvariant="normal">n</mi> </mrow> </msubsup> </mrow> </semantics></math>, for the imp–exp analyses, considering (<b>a</b>) discretization 1; (<b>b</b>) discretization 2; (<b>c</b>) discretization 3; and (<b>d</b>) discretization 4.</p> "> Figure 7 Cont.
<p>Computed values for (1) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </msubsup> </mrow> </semantics></math> and (2)<math display="inline"><semantics> <mrow> <mtext> </mtext> <msubsup> <mrow> <mi mathvariant="sans-serif">γ</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mi mathvariant="normal">n</mi> </mrow> </msubsup> </mrow> </semantics></math>, for the imp–exp analyses, considering (<b>a</b>) discretization 1; (<b>b</b>) discretization 2; (<b>c</b>) discretization 3; and (<b>d</b>) discretization 4.</p> "> Figure 8
<p>Computed values for (1) <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <msub> <mrow> <mi mathvariant="normal">t</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> </msub> </mrow> </semantics></math> and (2)<math display="inline"><semantics> <mrow> <mtext> </mtext> <mi mathvariant="sans-serif">Δ</mi> <msub> <mrow> <mi mathvariant="normal">t</mi> </mrow> <mrow> <mi mathvariant="normal">i</mi> </mrow> </msub> </mrow> </semantics></math> for the exp–exp analyses, considering (<b>a</b>) discretization 1; (<b>b</b>) discretization 2; (<b>c</b>) discretization 3; and (<b>d</b>) discretization 4.</p> "> Figure 9
<p>Time history results for <math display="inline"><semantics> <mrow> <mi mathvariant="normal">u</mi> </mrow> </semantics></math>, at a point located 10 m horizontally away from the applied source (discretization 4), considering solutions by (<b>a</b>) implicit and (<b>b</b>) explicit methods, as well as their hybrid extensions.</p> "> Figure 10
<p>Convergence curves for the discussed time-marching procedures and discretizations.</p> "> Figure 11
<p>Time–history results for the axial displacement at the middle of the rod, considering <math display="inline"><semantics> <mrow> <mrow> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mrow> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>: (<b>a</b>) implicit and (<b>b</b>) explicit approaches, as well as their hybrid extensions.</p> "> Figure 12
<p>Time–history results for the axial displacement at the middle of the rod, considering <math display="inline"><semantics> <mrow> <mrow> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mrow> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>: (<b>a</b>) implicit and (<b>b</b>) explicit approaches, as well as their hybrid extensions.</p> "> Figure 13
<p>Computed errors and CPU times for different material distributions, considering the exp–exp (open circle) and the imp–exp (solid circle, lighter gray color is depicted when a purely exp solution takes place, following the computed optimal <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mi mathvariant="normal">t</mi> </mrow> </semantics></math> value) approaches; <math display="inline"><semantics> <mrow> <mrow> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mrow> </mrow> </mrow> </semantics></math>.</p> "> Figure 14
<p>Computed errors and CPU times for different material distributions, considering the exp–exp (open circle) and the imp–exp (solid circle, lighter gray color is depicted when a purely exp solution takes place, following the computed optimal Δt value) approaches; <math display="inline"><semantics> <mrow> <mrow> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> </mrow> </mrow> </mrow> </semantics></math>.</p> "> Figure 15
<p>Computed errors and CPU times for different material distributions, considering the exp–exp (open circle) and the imp–exp (solid circle, lighter gray color is depicted when a purely exp solution takes place, following the computed optimal <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mi mathvariant="normal">t</mi> </mrow> </semantics></math> value) approaches; <math display="inline"><semantics> <mrow> <mrow> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> </mrow> </mrow> </mrow> </semantics></math>.</p> "> Figure 16
<p>Computed errors and CPU times for different material distributions, considering the exp–exp (open circle) and the imp–exp (solid circle, lighter gray color is depicted when a purely exp solution takes place, following the computed optimal <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mi mathvariant="normal">t</mi> </mrow> </semantics></math> value) approaches; <math display="inline"><semantics> <mrow> <mrow> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mrow> </mrow> </mrow> </semantics></math>.</p> "> Figure 17
<p>Computed errors and CPU times for different material distributions, considering the exp–exp (open circle) and the imp–exp (solid circle, lighter gray color is depicted when a purely exp solution takes place, following the computed optimal <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mi mathvariant="normal">t</mi> </mrow> </semantics></math> value) approaches; <math display="inline"><semantics> <mrow> <mrow> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>6</mn> </mrow> </mrow> </mrow> </semantics></math>.</p> "> Figure 18
<p>Computed errors and CPU times for different material distributions, considering the exp–exp (open circle) and the imp–exp (solid circle, lighter gray color is depicted when a purely exp solution takes place, following the computed optimal <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mi mathvariant="normal">t</mi> </mrow> </semantics></math> value) approaches; <math display="inline"><semantics> <mrow> <mrow> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>7</mn> </mrow> </mrow> </mrow> </semantics></math>.</p> "> Figure 19
<p>Computed errors and CPU times for different material distributions, considering the exp–exp (open circle) and the imp–exp (solid circle, lighter gray color is depicted when a purely exp solution takes place, following the computed optimal <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mi mathvariant="normal">t</mi> </mrow> </semantics></math> value) approaches; <math display="inline"><semantics> <mrow> <mrow> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>8</mn> </mrow> </mrow> </mrow> </semantics></math>.</p> "> Figure 20
<p>Analytical (black), imp–exp (light purple), and exp–exp (dark purple) time–history responses (all curves are visually the same) for the axial displacements at the middle of the rod, for <math display="inline"><semantics> <mrow> <mrow> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mrow> </mrow> </semantics></math> equals to (<b>a</b>) 2, (<b>b</b>) 3, (<b>c</b>) 4, (<b>d</b>) 5, (<b>e</b>) 6, (<b>f</b>) 7, and (<b>g</b>) 8; considering a percentage of material 2 equals to (1) 10%, (2) 50%, and (3) 90%.</p> "> Figure 20 Cont.
<p>Analytical (black), imp–exp (light purple), and exp–exp (dark purple) time–history responses (all curves are visually the same) for the axial displacements at the middle of the rod, for <math display="inline"><semantics> <mrow> <mrow> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <msub> <mrow> <mi mathvariant="normal">c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mrow> </mrow> </semantics></math> equals to (<b>a</b>) 2, (<b>b</b>) 3, (<b>c</b>) 4, (<b>d</b>) 5, (<b>e</b>) 6, (<b>f</b>) 7, and (<b>g</b>) 8; considering a percentage of material 2 equals to (1) 10%, (2) 50%, and (3) 90%.</p> "> Figure 21
<p>(<b>a</b>) Adopted spatial discretization for the homogeneous rod and its computed values for: (<b>b</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </msubsup> <mo>;</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">γ</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mi mathvariant="normal">n</mi> </mrow> </msubsup> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <msub> <mrow> <mi mathvariant="normal">t</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> </msub> </mrow> </semantics></math>; and (<b>e</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <msub> <mrow> <mi mathvariant="normal">t</mi> </mrow> <mrow> <mi mathvariant="normal">i</mi> </mrow> </msub> </mrow> </semantics></math>.</p> "> Figure 21 Cont.
<p>(<b>a</b>) Adopted spatial discretization for the homogeneous rod and its computed values for: (<b>b</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">Ω</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </msubsup> <mo>;</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">γ</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mi mathvariant="normal">n</mi> </mrow> </msubsup> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <msub> <mrow> <mi mathvariant="normal">t</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> </msub> </mrow> </semantics></math>; and (<b>e</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <msub> <mrow> <mi mathvariant="normal">t</mi> </mrow> <mrow> <mi mathvariant="normal">i</mi> </mrow> </msub> </mrow> </semantics></math>.</p> "> Figure 22
<p>Time–history results for the axial displacement at the middle of the homogeneous rod, considering (<b>a</b>) implicit and (<b>b</b>) explicit approaches, as well as their hybrid extensions.</p> "> Figure 23
<p>Geological models: (<b>a</b>) model 1—Buzios; (<b>b</b>) model 2—2DEW; and (<b>c</b>) model 3—2004BP.</p> "> Figure 24
<p>Subdomain divisions for the (1) imp–exp (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">γ</mi> </mrow> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mi mathvariant="normal">n</mi> </mrow> </msubsup> </mrow> </semantics></math> values are depicted) and (2) exp–exp (<math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <msub> <mrow> <mi mathvariant="normal">t</mi> </mrow> <mrow> <mi mathvariant="normal">i</mi> </mrow> </msub> </mrow> </semantics></math> values are depicted) methods, for (<b>a</b>) model 1; (<b>b</b>) model 2; and (<b>c</b>) model 3.</p> "> Figure 25
<p>Computed results along the discretized domain of model 1, for the (<b>a</b>) EG-α, (<b>b</b>) imp–exp, and (<b>c</b>) exp–exp methods at different time instants: (1) 3 s and (2) 6 s.</p> "> Figure 26
<p>Computed results along the discretized domain of model 2, for the (<b>a</b>) EG-α, (<b>b</b>) imp–exp, and (<b>c</b>) exp–exp methods at different time instants: (1) 3 s and (2) 6 s.</p> "> Figure 27
<p>Computed results along the discretized domain of model 3, for the (<b>a</b>) EG-α, (<b>b</b>) imp–exp, and (<b>c</b>) exp–exp methods at different time instants: (1) 10 s; (2) 15 s; and (3) 25 s.</p> ">
Abstract
:1. Introduction
2. Methods
2.1. Implicit–Explicit Approach
2.2. Explicit–Explicit Approach
2.3. Properties of the Methods
3. Results and Discussions
3.1. Theoretical Models
3.2. Applied Models
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Discretization | Method | Δt (10−2 s) | Error (10−1) | CPU Time (s) |
---|---|---|---|---|
1 | TR | 14.9256 (3.60) | 7.78 (2.08) | 19.6 (2.27) |
IG-α | 14.9256 (3.60) | 7.15 (1.91) | 19.8 (2.29) | |
IC | 14.9256 (3.60) | 7.31 (1.96) | 20.6 (2.38) | |
imp | 14.9256 (3.60) | 7.51 (2.01) | 19.5 (2.25) | |
CD | 4.60382 (1.11) | 7.98 (2.13) | 14.5 (1.67) | |
EG-α | 4.14927 (1.00) | 7.85 (2.10) | 15.9 (1.85) | |
EC | 8.62073 (2.08) | 7.91 (2.12) | 16.1 (1.86) | |
exp | 4.60382 (1.11) | 5.24 (1.40) | 14.0 (1.61) | |
imp–exp | 14.9256 (3.60) | 4.20 (1.12) | 9.45 (1.09) | |
exp–exp | 36.8306 (8.88) | 3.73 (1.00) | 8.67 (1.00) | |
2 | TR | 10.0516 (4.77) | 7.22 (2.76) | 32.7 (3.02) |
IG-α | 10.0516 (4.77) | 6.68 (2.25) | 32.6 (3.01) | |
IC | 10.0516 (4.77) | 6.81 (2.29) | 34.2 (3.17) | |
imp | 10.0516 (4.77) | 7.01 (2.36) | 32.8 (3.03) | |
CD | 2.33704 (1.11) | 7.62 (2.57) | 22.9 (2.12) | |
EG-α | 2.10629 (1.00) | 7.39 (2.49) | 24.2 (2.42) | |
EC | 4.37614 (2.08) | 7.51 (2.53) | 28.9 (2.67) | |
exp | 2.33704 (1.11) | 5.09 (1.71) | 22.7 (2.09) | |
imp–exp | 10.0516 (4.77) | 3.46 (1.17) | 11.2 (1.04) | |
exp–exp | 18.6963 (8.88) | 2.97 (1.00) | 10.8 (1.00) | |
3 | TR | 7.59321 (6.74) | 6.53 (3.08) | 101.4 (4.71) |
IG-α | 7.59321 (6.74) | 6.02 (2.84) | 101.9 (4.74) | |
IC | 7.59321 (6.74) | 6.12 (2.88) | 118.9 (5.53) | |
imp | 7.59321 (6.74) | 6.32 (2.98) | 103.6 (4.81) | |
CD | 1.24956 (1.11) | 7.03 (3.32) | 50.0 (2.33) | |
EG-α | 1.12619 (1.00) | 6.73 (3.17) | 57.4 (2.67) | |
EC | 2.33983 (2.08) | 6.92 (3.27) | 64.8 (3.01) | |
exp | 1.24956 (1.11) | 4.57 (2.15) | 50.2 (2.34) | |
imp–exp | 7.59321 (6.74) | 2.60 (1.23) | 21.5 (1.00) | |
exp–exp | 19.9930 (17.8) | 2.12 (1.00) | 26.4 (1.23) | |
4 | TR | 6.20276 (8.48) | 6.17 (3.55) | 159.9 (3.85) |
IG-α | 6.20276 (8.48) | 5.60 (3.22) | 160.2 (3.86) | |
IC | 6.20276 (8.48) | 5.71 (3.29) | 174.1 (4.19) | |
imp | 6.20276 (8.48) | 5.93 (3.41) | 160.2 (3.14) | |
CD | 0.81175 (1.11) | 6.66 (3.84) | 88.8 (2.14) | |
EG-α | 0.73160 (1.00) | 6.38 (3.67) | 114.9 (2.77) | |
EC | 1.52001 (2.08) | 6.59 (3.79) | 126.7 (3.05) | |
exp | 0.81174 (1.11) | 4.18 (3.40) | 89.0 (2.15) | |
imp–exp | 6.20276 (8.48) | 2.19 (1.26) | 41.5 (1.00) | |
exp–exp | 12.9880 (17.8) | 1.74 (1.00) | 49.5 (1.19) |
Method | CPU Time (s) | |||
---|---|---|---|---|
4 | TR | 0.62499 (1.11) | 7.96 (9.68) | 121.4 (8.70) |
IG-α | 0.62499 (1.11) | 7.56 (9.19) | 120.1 (8.61) | |
IC | 0.62499 (1.11) | 8.36 (10.1) | 230.4 (16.5) | |
imp | 0.62499 (1.11) | 5.77 (7.01) | 120.6 (8.64) | |
CD | 0.62499 (1.11) | 5.84 (7.10) | 16.6 (1.19) | |
EG-α | 0.56317 (1.00) | 6.20 (7.53) | 18.3 (1.31) | |
EC | 1.17046 (2.08) | 6.24 (7.58) | 22.4 (1.60) | |
exp | 0.62499 (1.11) | 5.75 (6.98) | 16.6 (1.19) | |
imp–exp | 0.62499 (1.11) | 5.75 (6.98) | 16.6 (1.19) | |
exp–exp | 2.49998 (4.44) | 0.82 (1.00) | 13.9 (1.00) | |
6 | TR | 2.49998 (6.66) | 18.87 (3.94) | 30.43 (1.84) |
IG-α | 2.49998 (6.66) | 22.23 (4.64) | 31.52 (1.91) | |
IC | 2.49998 (6.66) | 18.99 (3.96) | 56.68 (3.43) | |
imp | 2.49998 (6.66) | 18.50 (3.86) | 30.35 (1.83) | |
CD | 0.41666 (1.11) | 8.49 (1.77) | 20.65 (1.25) | |
EG-α | 0.37545 (1.00) | 8.70 (1.82) | 25.29 (1.53) | |
EC | 0.78030 (2.08) | 8.78 (1.83) | 28.25 (1.71) | |
exp | 0.41666 (1.11) | 8.41 (1.76) | 20.78 (1.26) | |
imp–exp | 2.49998 (6.66) | 8.93 (1.86) | 16.56 (1.00) | |
exp–exp | 1.66664 (4.44) | 4.79 (1.00) | 16.54 (1.00) |
Method | CPU Time (s) | ||
---|---|---|---|
TR | 1.42924 (1.84) | 8.65 (1.67) | 28.8 (4.97) |
IG-α | 1.42924 (1.84) | 8.34 (1.61) | 29.9 (5.15) |
IC | 1.42924 (1.84) | 8.49 (1.63) | 53.1 (9.15) |
imp | 1.42924 (1.84) | 8.29 (1.60) | 28.6 (4.92) |
CD | 0.86030 (1.11) | 7.42 (1.43) | 8.2 (1.41) |
EG-α | 0.77536 (1.00) | 8.01 (1.54) | 9.4 (1.61) |
EC | 1.61092 (2.08) | 7.43 (1.43) | 11.0 (1.89) |
exp | 0.86090 (1.11) | 6.41 (1.23) | 8.3 (1.42) |
imp–exp | 1.42924 (1.84) | 5.18 (1.00) | 6.4 (1.11) |
exp–exp | 3.44120 (4.44) | 6.19 (1.19) | 5.8 (1.00) |
Model | Method | CPU Time (s) | |
---|---|---|---|
1 | CD | 0.64051 (1.11) | 5233.7 (1.68) |
EG-α | 0.57727 (1.00) | 5462.8 (1.75) | |
EC | 1.19936 (2.08) | 5683.6 (1.83) | |
exp | 0.64051 (1.11) | 5261.2 (1.69) | |
imp–exp | 0.86751 (1.50) | 4282.5 (1.38) | |
exp–exp | 2.56204 (4.44) | 3112.9 (1.00) | |
2 | CD | 0.24255 (1.11) | 8661.5 (1.99) |
EG-α | 0.21861 (1.00) | 8991.6 (2.06) | |
EC | 0.45419 (2.08) | 9511.1 (2.18) | |
exp | 0.24255 (1.11) | 8654.5 (1.98) | |
imp–exp | 0.34235 (1.57) | 5436.3 (1.25) | |
exp–exp | 1.94044 (8.88) | 4362.2 (1.00) | |
3 | CD | 0.32415 (1.11) | 11322.8 (2.15) |
EG-α | 0.29215 (1.00) | 12283.3 (2.34) | |
EC | 0.60698 (2.08) | 14773.1 (2.81) | |
exp | 0.32415 (1.11) | 11737.7 (2.23) | |
imp–exp | 0.42489 (1.45) | 5259.40 (1.00) | |
exp–exp | 10.3729 (35.5) | 5883.30 (1.11) |
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Soares, D., Jr.; Sales, I.d.S.; Pinto, L.R.; Mansur, W.J. A Study on Adaptive Implicit–Explicit and Explicit–Explicit Time Integration Procedures for Wave Propagation Analyses. Acoustics 2024, 6, 651-680. https://doi.org/10.3390/acoustics6030036
Soares D Jr., Sales IdS, Pinto LR, Mansur WJ. A Study on Adaptive Implicit–Explicit and Explicit–Explicit Time Integration Procedures for Wave Propagation Analyses. Acoustics. 2024; 6(3):651-680. https://doi.org/10.3390/acoustics6030036
Chicago/Turabian StyleSoares, Delfim, Jr., Isabelle de Souza Sales, Lucas Ruffo Pinto, and Webe João Mansur. 2024. "A Study on Adaptive Implicit–Explicit and Explicit–Explicit Time Integration Procedures for Wave Propagation Analyses" Acoustics 6, no. 3: 651-680. https://doi.org/10.3390/acoustics6030036