System and method for identifying optimal locations for strain gage placement on a structure
A method for identifying optimal locations for strain gage placement on a structure includes: providing a finite element (FE) model of the structure with an initial number I of candidate strain gage locations; conducting FE analysis to produce a strain matrix [S] of strain predictions, based on a load matrix [L] of external loads being applied to the FE model; relating the matrices to each other in a linear relationship model [L]=[S][β] having a residual error function; and reducing the initial number of the candidate locations to a reduced number by using a feature selection algorithm which minimizes the residual error function until a predetermined fidelity is achieved, resulting in a subset of the coefficients in the coefficients matrix [β] being zero and a remainder of the coefficients being non-zero, wherein the non-zero coefficients correspond to respective strain matrix columns and to the reduced number of candidate locations.
1 . A method comprising:
providing one or more strain gages;
providing a structure;
providing a finite element model of the structure, wherein the finite element model includes a mesh of elements that correspond with an initial number I of respective candidate strain gage locations on or within the structure;
conducting finite element analysis on the finite element model to produce a strain matrix [S] of strain predictions having m rows and M strain matrix columns, based on a load matrix [L] of external loads having n rows and N load matrix columns being applied to selected elements in the finite element model, wherein each of the strain matrix columns corresponds to a respective one of the candidate strain gage locations and each of the load matrix columns corresponds to a respective one of the external loads;
relating the strain and load matrices to each other in a linear relationship model [L]=[S][β] having a residual error function ƒ(L−Sβ), where [β] is a matrix of coefficients having b rows and B columns, and wherein each of the b rows of coefficients corresponds to a respective one of the M strain matrix columns, and wherein L, S and β represent external loads, strain predictions and coefficients, respectively, which correspond to each other according to the linear relationship model [L]=[S][β];
applying a feature selection algorithm to the linear relationship model [L]=[S][β] to reduce the initial number I of the candidate strain gage locations to a reduced number R of the candidate strain gage locations, wherein the feature selection algorithm minimizes the residual error function ƒ(L−Sβ) until a predetermined level of fidelity is achieved, resulting in a solution set in which a subset of the coefficients in the coefficients matrix are zero and a remainder of the coefficients are non-zero, wherein the non-zero coefficients correspond to respective ones of the strain matrix columns and to the reduced number R of the candidate strain gage locations; and
placing the one or more strain gages on the structure at one or more of the reduced number R of the candidate strain gage locations.
2 . The method of claim 1 , further comprising:
selecting at least some of the reduced number R of candidate strain gage locations as intended locations for strain gage placement.
3 . The method of claim 2 , further comprising:
placing a respective strain gage at each of the intended locations.
4 . The method of claim 3 , further comprising:
receiving a respective output signal from each respective strain gage.
5 . The method of claim 1 , wherein the feature selection algorithm is a least absolute shrinkage and selection (LASSO) regression algorithm, a multitask LASSO regression algorithm or an adaptive LASSO regression algorithm.
6 . The method of claim 1 , wherein the feature selection algorithm is an elastic net regression algorithm, a forward stepwise regression algorithm or a backward stepwise regression algorithm.
7 . The method of claim 1 , wherein the strain and load matrices have the same number of rows as each other and the number of rows b of coefficients in the [β] matrix is equal to the number M of strain matrix columns.
8 . The method of claim 7 , wherein the strain and load matrices each have n rows, and where each i th set of rows in the strain and load matrices together represent a unique set of load conditions, where i is a number from 1 to n.
9 . The method of claim 1 , wherein the step of applying the feature selection algorithm continues until the initial number I of the candidate strain gage locations is reduced to less than or equal to a predetermined maximum number.
10 . The method of claim 1 , wherein the step of applying the feature selection algorithm continues until the residual error function ƒ(L−Sβ) is less than or equal to a predetermined maximum allowable amount.
11 . A method for identifying optimal locations for strain gage placement on a structure, comprising:
providing one or more strain gages;
providing a structure;
providing a finite element model of the structure, wherein the finite element model includes a mesh of elements that correspond with an initial number I of respective candidate strain gage locations on or within the structure;
conducting finite element analysis on the finite element model to produce a strain matrix [S] of strain predictions having m rows and M strain matrix columns, based on a load matrix [L] of external loads having n rows and N load matrix columns being applied to selected elements in the finite element model, wherein each of the strain matrix columns corresponds to a respective one of the candidate strain gage locations and each of the load matrix columns corresponds to a respective one of the external loads;
relating the strain and load matrices to each other in a linear relationship model [L]=[S][β] having a residual error function ƒ(L−Sβ), where [β] is a matrix of coefficients having b rows and B columns, and wherein each of the b rows of coefficients corresponds to a respective one of the M strain matrix columns, and wherein L, S and β represent external loads, strain predictions and coefficients, respectively, which correspond to each other according to the linear relationship model [L]=[S][β];
reducing the initial number I of the candidate strain gage locations to a reduced number R of the candidate strain gage locations by using a multitask least absolute shrinkage and selection (LASSO) regression algorithm operating on the linear relationship model [L]=[S][β], wherein the multitask LASSO regression algorithm minimizes the residual error function ƒ(L−Sβ) subject to a penalty term, resulting in a solution set in which a subset of the coefficients in the coefficients matrix are zero and a remainder of the coefficients are non-zero, wherein the non-zero coefficients correspond to respective ones of the strain matrix columns and to the reduced number R of the candidate strain gage locations;
selecting at least some of the reduced number R of candidate strain gage locations as intended locations for strain gage placement;
placing a respective strain gage at each of the intended locations; and
receiving a respective output signal from each respective strain gage.
12 . The method of claim 11 , wherein the strain and load matrices have the same number of rows as each other and the number of rows b of coefficients in the [β] matrix is equal to the number M of strain matrix columns.
13 . The method of claim 12 , wherein the strain and load matrices each have n rows, and where each i th set of rows in the strain and load matrices together represent a unique set of load conditions, where i is a number from 1 to n.
14 . The method of claim 11 , wherein the step of reducing the initial number I of the candidate strain gage locations to a reduced number R by using the multitask LASSO regression algorithm continues until the initial number I of the candidate strain gage locations is reduced to less than or equal to a predetermined maximum number.
15 . The method of claim 11 , wherein the step of reducing the initial number I of the candidate strain gage locations to a reduced number R by using the multitask LASSO regression algorithm continues until the residual error function ƒ(L−Sβ) subject to the penalty term is less than or equal to a predetermined maximum allowable amount.
16 . A system comprising:
a structure;
one or more strain gages; and
hardware comprising:
a finite element module for providing a finite element model of the structure, wherein the finite element model includes a mesh of elements that correspond with an initial number I of respective candidate strain gage locations on or within the structure;
an analysis module, operatively connected with the finite element module, for conducting finite element analysis on the finite element model to produce a strain matrix [S] of strain predictions having m rows and M strain matrix columns, based on a load matrix [L] of external loads having n rows and N load matrix columns being applied to selected elements in the finite element model, wherein each of the strain matrix columns corresponds to a respective one of the candidate strain gage locations and each of the load matrix columns corresponds to a respective one of the external loads;
a relationship module, operatively connected with the analysis module, for relating the strain and load matrices to each other in a linear relationship model [L]=[S][β] having a residual error function ƒ(L−Sβ), where [β] is a matrix of coefficients having b rows and B columns, and wherein each of the b rows of coefficients corresponds to a respective one of the M strain matrix columns, and wherein L, S and β represent external loads, strain predictions and coefficients, respectively, which correspond to each other according to the linear relationship model [L]=[S][β]; and
a reduction module, operatively connected with the relationship module, for reducing the initial number I of the candidate strain gage locations to a reduced number R of the candidate strain gage locations by applying a feature selection algorithm to the linear relationship model [L]=[S][β], wherein the feature selection algorithm minimizes the residual error function ƒ(L−Sβ) until a predetermined level of fidelity is achieved, resulting in a solution set in which a subset of the coefficients in the coefficients matrix are zero and a remainder of the coefficients are non-zero, wherein the non-zero coefficients correspond to respective ones of the strain matrix columns and to the reduced number R of the candidate strain gage locations,
wherein the one or more strain gages are placed at one or more of the reduced number R of the candidate strain gage locations.
17 . The system of claim 16 , wherein the hardware further comprises:
a selection module, operatively connected with the reduction module, for selecting at least some of the reduced number R of candidate strain gage locations as intended locations for strain gage placement.
18 . The system of claim 16 , wherein the feature selection algorithm is a least absolute shrinkage and selection (LASSO) regression algorithm, a multitask LASSO regression algorithm, an adaptive LASSO regression algorithm, an elastic net regression algorithm, a forward stepwise regression algorithm or a backward stepwise regression algorithm.
19 . The system of claim 16 , wherein the strain and load matrices each have n rows, and where each i th set of rows in the strain and load matrices together represent a unique set of load conditions, where i is a number from 1 to n.
20 . The system of claim 16 , wherein the applying of the feature selection algorithm continues until the initial number I of the candidate strain gage locations is reduced to less than or equal to a predetermined maximum number or until the residual error function ƒ(L−Sβ) is less than or equal to a predetermined maximum allowable amount.