search.noResults

search.searching

dataCollection.invalidEmail
note.createNoteMessage

search.noResults

search.searching

orderForm.title

orderForm.productCode
orderForm.description
orderForm.quantity
orderForm.itemPrice
orderForm.price
orderForm.totalPrice
orderForm.deliveryDetails.billingAddress
orderForm.deliveryDetails.deliveryAddress
orderForm.noItems
OPTIMIZATION ALGORITHM


1. Create a pool of node points located exterior of the prob- lem domain. This set should be in no particular pattern. 


2. Create a pool of collocation points located on the problem boundary. These are locations where the potential is known.


3. Create a one node approximation function for each com- bination of node and collocation point. Record the error for each.


4. Select the node and collocation pair that produced the least error.


5. Create a two node model, utilizing the selected pair from step 4 as one of the node pairs.


6. Test all two node approximations for error. The approxi- mation function that results in the least error becomes the best two node model.


7. Repeat steps 4-6 until the number of nodes desired in the model is reached.


The following test problems are examined to demonstrate the validity of the algorithm.


Example Problem: Pressure Source


To demonstrate the algorithm, a concerning soil-water pres- sure source, such as a longitudinal crack along the surface of a high pressure water pipeline, leads to detailed analysis, including forensic as well as remediation examination, involv- ing complex computational modeling methods. The pipeline exerts pressure uniformly and can be modeled by the equation,


Figure 1 - Problem Domain


To create the space of candidate node loca- tions for use in the basis    adequate amount of nodes must be assessed. To minimize the algo- rithm’s run time, the number of node loca- tions examined is lim- ited to 512. Figure 2 depicts the location of each of the candidate nodes.





source location (xj,yj,zj by


where k is a real-valued coefficient, and


where (xk,yk,zk) is the kth node. To solve for the k’s, pressures must Pl =(xl,yl,zl), points on the 


Pl. Set k = l 


           


 


 length = 8, depthheight h


 





Similar to the creation of the nodes, candi- date collocation point locations must be posi- tioned on the problem boundary. The number of collocation points need not be the same as the number of candi- date node locations. For this example, there will be 1000 candidate collo- cation points. Figure 3 depicts the distribution of candidate collocation point locations.


Figure 2 - Candidate node locations assessed.


Figure 3 - Candidate collocation points in the problem domain.


The accuracy of each one node approximation function must be evaluated and compared. Let n = the number of candidate nodes and m = the number of candidate collocation points. To test accuracy, every combination of candidate nodes and candidate collocation points will be paired and used to create an approximation function. Thus, there will be n x m approxi- mation functions to be compared for computational error. The ordered pair with the least error is deemed the optimized node and collocation point location and combination for use in a one node model. Table 1 demonstrates the comparison of errors that occurs automatically within the algorithm for the one test node model and 5 test collocation locations.


Because the number of possible combinations of nodes and collocation points for the sample size that is used is large  into the process that is occurring. Table 1 lists the possible -


8 TPG • Jan.Feb.Mar 2019 www.aipg.org


Page 1  |  Page 2  |  Page 3  |  Page 4  |  Page 5  |  Page 6  |  Page 7  |  Page 8  |  Page 9  |  Page 10  |  Page 11  |  Page 12  |  Page 13  |  Page 14  |  Page 15  |  Page 16  |  Page 17  |  Page 18  |  Page 19  |  Page 20  |  Page 21  |  Page 22  |  Page 23  |  Page 24  |  Page 25  |  Page 26  |  Page 27  |  Page 28  |  Page 29  |  Page 30  |  Page 31  |  Page 32  |  Page 33  |  Page 34  |  Page 35  |  Page 36  |  Page 37  |  Page 38  |  Page 39  |  Page 40  |  Page 41  |  Page 42  |  Page 43  |  Page 44  |  Page 45  |  Page 46  |  Page 47  |  Page 48  |  Page 49  |  Page 50  |  Page 51  |  Page 52  |  Page 53  |  Page 54  |  Page 55  |  Page 56  |  Page 57  |  Page 58  |  Page 59  |  Page 60  |  Page 61  |  Page 62  |  Page 63  |  Page 64