Stress Responses for Topology and Free-Size Optimization

The following approaches are available to handle stress constraints for Topology and Free-size Optimization.

Note: Augmented Lagrange Method is only supported for Topology Optimization.

Norm-based Approach (Topology and Free-size Optimization)

The Norm-based approach is the default method for handling stress responses for Topology and Free-Size Optimization. This method is used when corresponding stress response RTYPE’s on the DRESP1 Bulk Data Entry are input.

The Response-NORM aggregation is internally used to calculate the stress responses for groups of elements in the model. Solid, shell, bar stresses and solid corner stresses are supported with the Response-Norm aggregation approach (Free-size Optimization is only supported for shell stresses). Refer to NORM Method for more information.

Augmented Lagrange Method (ALM) (Topology Optimization only)

The Augmented Lagrange Method (ALM) is an alternative method for handling stress responses for Topology Optimization. It can be activated using DOPTPRM,ALMTOSTR,1 when DRESP1 Bulk Data Entry is used to specify local stress responses.

ALM is also an alternative method to efficiently solve topology optimization problems with local stress constraints, which is stated in Equation 1.

min f ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyBaiaacM gacaGGUbGaamOzamaabmaabaGaamiwaaGaayjkaiaawMcaaaaa@3BF9@ (1)
s . t . g j ( X ) 0 , j = 1 , , m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiaac6 cacaWG0bGaaiOlaiaaysW7caWGNbWaaSbaaSqaaiaadQgaaeqaaOWa aeWaaeaacaWGybaacaGLOaGaayzkaaGaeyizImQaaGimaiaacYcaca WGQbGaeyypa0JaaGymaiaacYcacaaMc8UaeSOjGSKaaiilaiaaykW7 caaMe8UaamyBamaaBaaaleaacaaIXaaabeaaaaa@4DFC@

g j ( X ) 0 , j = m 1 + 1 , , m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGQbaabeaakmaabmaabaGaamiwaaGaayjkaiaawMcaaiab gsMiJkaaicdacaGGSaGaamOAaiabg2da9iaad2gadaWgaaWcbaGaaG ymaaqabaGccqGHRaWkcaaIXaGaaiilaiaaykW7cqWIMaYscaGGSaGa aGPaVlaaysW7caWGTbWaaSbaaSqaaiaaigdaaeqaaaaa@4BDF@

Where,
X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaaaa@36B3@
Vector of topology design variables
f ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWGybGaaiykaaaa@38F7@
Objective function
g j ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGQbaabeaakmaabmaabaGaamiwaaGaayjkaiaawMcaaaaa @3A4D@
jth constraint
m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaaIXaaabeaaaaa@37AF@
Number of local stress constraints
m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36C8@
Total number of constraints
Typically, m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaaIXaaabeaaaaa@37AF@ is equivalent to the number of elements and is a large number. ALM converts the optimization Equation 1 to the following format:(2)
min f ( X ) + 1 m 1 j = 1 m 1 [ λ j h j ( X ) + μ 2 h j 2 ( X ) ]   ,   j = 1 , , m 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaciyBaiaacMgacaGGUbGaamOzamaabmaapaqaa8qacaWGybaacaGL OaGaayzkaaGaey4kaSYaaSaaa8aabaWdbiaaigdaa8aabaWdbiaad2 gapaWaaSbaaSqaa8qacaaIXaaapaqabaaaaOWdbmaawahabeWcpaqa a8qacaWGQbGaeyypa0JaaGymaaWdaeaapeGaamyBa8aadaWgaaadba Wdbiaaigdaa8aabeaaa0qaa8qacqGHris5aaGcdaWadaWdaeaapeGa eq4UdW2damaaBaaaleaapeGaamOAaaWdaeqaaOWdbiaadIgapaWaaS baaSqaa8qacaWGQbaapaqabaGcpeWaaeWaa8aabaWdbiaadIfaaiaa wIcacaGLPaaacqGHRaWkdaWcaaWdaeaapeGaeqiVd0gapaqaa8qaca aIYaaaaiaadIgapaWaa0baaSqaa8qacaWGQbaapaqaa8qacaaIYaaa aOWaaeWaa8aabaWdbiaadIfaaiaawIcacaGLPaaaaiaawUfacaGLDb aacaGGGcGaaiilaiaacckacaWGQbGaeyypa0JaaGymaiaacYcacqGH MacVcaGGSaGaamyBa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@6582@
(3)EQ 2
s . t . g j ( X ) 0 , j = m 1 + 1 , , m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiaac6 cacaWG0bGaaiOlaiaaysW7caWGNbWaaSbaaSqaaiaadQgaaeqaaOWa aeWaaeaacaWGybaacaGLOaGaayzkaaGaeyizImQaaGimaiaacYcaca WGQbGaeyypa0JaamyBamaaBaaaleaacaaIXaaabeaakiabgUcaRiaa igdacaGGSaGaaGPaVlablAciljaacYcacaaMc8UaaGjbVlaad2gada WgaaWcbaGaaGymaaqabaaaaa@50C1@
Where,
h j ( X ) = m a x [ g j ( X ) , λ j μ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiAa8aadaWgaaWcbaWdbiaadQgaa8aabeaak8qadaqadaWdaeaa peGaamiwaaGaayjkaiaawMcaaiabg2da9iaad2gacaWGHbGaamiEam aadmaapaqaa8qacaWGNbWdamaaBaaaleaapeGaamOAaaWdaeqaaOWd bmaabmaapaqaa8qacaWGybaacaGLOaGaayzkaaGaaiilaiabgkHiTm aalaaapaqaa8qacqaH7oaBpaWaaSbaaSqaa8qacaWGQbaapaqabaaa keaapeGaeqiVd0gaaaGaay5waiaaw2faaaaa@4C3A@
g j ( X ) = E j ( σ j σ l i m 1 ) [ 1 + ( σ j σ l i m 1 ) 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4za8aadaWgaaWcbaWdbiaadQgaa8aabeaak8qadaqadaWdaeaa peGaamiwaaGaayjkaiaawMcaaiabg2da9iaadweapaWaaSbaaSqaa8 qacaWGQbaapaqabaGcpeWaaeWaa8aabaWdbmaalaaapaqaa8qacqaH dpWCpaWaaSbaaSqaa8qacaWGQbaapaqabaaakeaapeGaeq4Wdm3dam aaBaaaleaapeGaamiBaiaadMgacaWGTbaapaqabaaaaOWdbiabgkHi TiaaigdaaiaawIcacaGLPaaadaWadaWdaeaapeGaaGymaiabgUcaRm aabmaapaqaa8qadaWcaaWdaeaapeGaeq4Wdm3damaaBaaaleaapeGa amOAaaWdaeqaaaGcbaWdbiabeo8aZ9aadaWgaaWcbaWdbiaadYgaca WGPbGaamyBaaWdaeqaaaaak8qacqGHsislcaaIXaaacaGLOaGaayzk aaWdamaaCaaaleqabaWdbiaaikdaaaaakiaawUfacaGLDbaaaaa@5A2D@
,   j = 1 , , m 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaiilaiaacckacaWGQbGaeyypa0JaaGymaiaacYcacqGHMacVcaGG SaGaamyBa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@3F85@
E j =ε+( 1ε ) [ tanh( βη )+tanh( β( ρ j η ) ) tanh( βη )+tanh( β( 1η ) ) ] p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyra8aadaWgaaWcbaWdbiaadQgaa8aabeaak8qacqGH9aqpcqaH 1oqzcqGHRaWkdaqadaWdaeaapeGaaGymaiabgkHiTiabew7aLbGaay jkaiaawMcaamaadmaapaqaa8qadaWcaaWdaeaapeGaamiDaiaadgga caWGUbGaamiAamaabmaapaqaa8qacqaHYoGycqaH3oaAaiaawIcaca GLPaaacqGHRaWkcaWG0bGaamyyaiaad6gacaWGObWaaeWaa8aabaWd biabek7aInaabmaapaqaa8qacqaHbpGCpaWaaSbaaSqaa8qacaWGQb aapaqabaGcpeGaeyOeI0Iaeq4TdGgacaGLOaGaayzkaaaacaGLOaGa ayzkaaaapaqaa8qacaWG0bGaamyyaiaad6gacaWGObWaaeWaa8aaba Wdbiabek7aIjabeE7aObGaayjkaiaawMcaaiabgUcaRiaadshacaWG HbGaamOBaiaadIgadaqadaWdaeaapeGaeqOSdi2aaeWaa8aabaWdbi aaigdacqGHsislcqaH3oaAaiaawIcacaGLPaaaaiaawIcacaGLPaaa aaaacaGLBbGaayzxaaWdamaaCaaaleqabaWdbiaadchaaaaaaa@71FB@
ρ j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS baaSqaaiaadQgaaeqaaaaa@38B1@
Element density
σ j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadQgaaeqaaaaa@38B4@
Element von Mises stress
σ lim MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiGacYgacaGGPbGaaiyBaaqabaaaaa@3A95@
stress upper bound
λ j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS baaSqaaiaadQgaaeqaaaaa@38A5@
Lagrange multiplier estimator
μ > 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey Opa4JaaGimaaaa@394E@
Quadratic penalty factor
The penalty factor is typically updated as:(4)
μ ( k + 1 ) = m i n ( α μ ( k ) , μ m a x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiVd02damaaCaaaleqabaWdbmaabmaapaqaa8qacaWGRbGaey4k aSIaaGymaaGaayjkaiaawMcaaaaakiabg2da9iaad2gacaWGPbGaam OBamaabmaapaqaa8qacqaHXoqycqaH8oqBpaWaaWbaaSqabeaapeWa aeWaa8aabaWdbiaadUgaaiaawIcacaGLPaaaaaGccaGGSaGaeqiVd0 2damaaBaaaleaapeGaamyBaiaadggacaWG4baapaqabaaak8qacaGL OaGaayzkaaaaaa@4DC0@
Where,
α > 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey Opa4JaaGymaaaa@3938@
Update parameter
μ max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaaaaa@3A8C@
Upper limit to prevent numerical instabilities

The Lagrange multiplier estimators are updated as λ j ( k + 1 ) = λ j ( k ) + μ ( k ) h j ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4UdW2damaaDaaaleaapeGaamOAaaWdaeaapeWaaeWaa8aabaWd biaadUgacqGHRaWkcaaIXaaacaGLOaGaayzkaaaaaOGaeyypa0Jaeq 4UdW2damaaDaaaleaapeGaamOAaaWdaeaapeWaaeWaa8aabaWdbiaa dUgaaiaawIcacaGLPaaaaaGccqGHRaWkcqaH8oqBpaWaaWbaaSqabe aapeWaaeWaa8aabaWdbiaadUgaaiaawIcacaGLPaaaaaGccaWGObWd amaaBaaaleaapeGaamOAaaWdaeqaaOWdbmaabmaapaqaa8qacaWGyb aacaGLOaGaayzkaaaaaa@4E6A@ .

In general, the number of local stress constraints m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaaIXaaabeaaaaa@37AF@ is very large. Directly solving Equation 1 is computationally time consuming. By penalizing the stress constraints onto the objective function, the total number of constraints can be significantly reduced. As a result, the optimization problem can be efficiently solved.

The parameters are set as μ ( 0 ) = 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaW baaSqabeaacaGGOaGaaGimaiaacMcaaaGccqGH9aqpcaaIXaGaaGim aaaa@3C51@ , μ max = 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaGccqGH9aqpcaaIXaGaaGim amaaCaaaleqabaGaaGOnaaaaaaa@3DFE@ , α = 1.3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey ypa0JaaGymaiaac6cacaaIZaaaaa@3AA5@ , η = 0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdGMaey ypa0JaaGimaiaac6cacaaI1aaaaa@3AB3@ , ε = 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaey ypa0JaaGymaiaaicdadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3BD2@ , p = 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9iaaiodaaaa@388E@ . β is initially set to 1.0 and multiplied by 1.3 in every five iterations, with an upper limit of 30.0. Based on this process, Topology Optimization is carried out within one phase.
Note: Except for this ALM, OptiStruct generally uses a multi-phase strategy to solve topology optimization problems.

The default Stress-norm (P-norm) method continues to efficiently solve Topology Optimization problems with local stress constraints. The ALM 1 is a good alternative for such models.

Global von Mises Stress Response (Topology and Free-size Optimization)

The von Mises stress constraints may be defined for topology and free-size optimization through the STRESS optional continuation line on the DTPL or the DSIZE card. There are a number of restrictions with this constraint:
  • The definition of stress constraints is limited to a single von Mises permissible stress. The phenomenon of singular topology is pronounced when different materials with different permissible stresses exist in a structure. Singular topology refers to the problem associated with the conditional nature of stress constraints, that is, the stress constraint of an element disappears when the element vanishes. This creates another problem in that a huge number of reduced problems exist with solutions that cannot usually be found by a gradient-based optimizer in the full design space.
  • Stress constraints for a partial domain of the structure are not allowed because they often create an ill-posed optimization problem since elimination of the partial domain would remove all stress constraints. Consequently, the stress constraint applies to the entire model when active, including both design and non-design regions, and stress constraint settings must be identical for all DSIZE and DTPL cards.
  • The capability has built-in intelligence to filter out artificial stress concentrations around point loads and point boundary conditions. Stress concentrations due to boundary geometry are also filtered to some extent as they can be improved more effectively with local shape optimization.
  • Due to the large number of elements with active stress constraints, no element stress report is given in the table of retained constraints in the .out file. The iterative history of the stress state of the model can be viewed in HyperView or HyperMesh.
  • Stress constraints do not apply to 1D elements.
  • Stress constraints may not be used when enforced displacements are present in the model.

The buckling factor can be constrained for shell topology optimization problems with a base thickness not equal to zero. Constraints on the buckling factor are not allowed in any other cases of topology optimization.

The following responses are currently available as the objective or as constraint functions for elements that do not form part of the design space:
Composite Stress Composite Strain Composite Failure Criterion
Frequency Response Stress Frequency Response Strain Frequency Response Force